∎

11institutetext: C. Karcher 22institutetext: Department of Aeronautics and
Astronautics, Massachusetts Institute of Technology
77 Massachusetts Avenue, Cambridge, MA 02139, USA
22email<EMAIL_ADDRESS>

# Data Fitting with Signomial Programming Compatible Difference of Convex
Functions

Cody Karcher

(Received: date / Accepted: date)

###### Abstract

Signomial Programming (SP) has proven to be a powerful tool for engineering
design optimization, striking a balance between the computational efficiency
of Geometric Programming (GP) and the extensibility of more general
optimization methods like Sequential Quadratic Programming (SQP). But when an
existing engineering analysis tool is incompatible with the mathematics of the
SP formulation, options are limited. Previous literature has suggested schemes
for fitting GP compatible models to pre-computed data, but no methods have yet
been proposed that take advantage of the increased modeling flexibility
available in SP. This paper describes a new Soft Difference of Max Affine
(SDMA) function class that is constructed from existing methods of GP
compatible fitting and the theory of Difference of Convex (DC) functions. When
a SDMA function is fit to data in log-log transformed space, it becomes either
a signomial or a set of signomials upon inverse transformation. Three examples
of fitting are presented here, including simple test cases in 2D and 3D, and a
fit to the performance data of the NACA 24xx family of airfoils. In each case,
RMS error is driven to less than 1%.

###### Keywords:

Data Fitting Signomial Programming Log Convexity Difference of Convex
Functions

##  1 Motivation

Geometric Programming (GP) and Signomial Programming (SP) are two related
classes of non-linear optimization formulations that have recently been
applied, with great success, to the field of aircraft design
hoburg2014geometric ; torenbeek2013advanced ; hoburg2013fast ;
kirschen2016signomial ; brown2018vehicle ; york2018efficient ; burton2018solar
; lin2020simultaneous ; kirschen2018application ; york2018turbofan ;
saab2018robust ; hall2018assessment . Despite these successes, the GP and SP
formulations do not allow the use of black box analysis tools that are common
in practical design problems martins2013multidisciplinary . In aircraft
design, these tools include high fidelity analyses such as Computational Fluid
Dynamics (CFD) and Finite Element Analysis (FEA), and low fidelity analyses
like XFOIL drela1989xfoil  and AVL drela2010avl . Previous work by Hoburg et.
al. hoburg2016data  addressed this issue for Geometric Programming by
proposing three methods for fitting a GP compatible function to a pre-computed
dataset, but the additional modeling flexibility available in Signomial
Programming goes untreated. This work fills this gap by proposing a method for
fitting a SP compatible function to data in cases where the Hoburg methods
hoburg2016data  lack sufficient accuracy.

##  2 Geometric Programming and Signomial Programming

###  2.1 Geometric Programming

Geometric Programs are built upon two fundamental building blocks: monomial
and posynomial functions. A monomial function is defined as the product of a
leading constant with each variable raised to a real power boyd2007tutorial :

$m(\textbf{x})=c{x_{1}}^{a_{1}}{x_{2}}^{a_{2}}...\;{x_{n}}^{a_{n}}=c\prod_{i=1}^{N}x_{i}^{a_{i}}$
(1)

A posynomial is simply the sum of monomials boyd2007tutorial , which can be
defined in notation as:

$p(\textbf{x})=m_{1}(\textbf{x})+m_{2}(\textbf{x})+...+m_{n}(\textbf{x})=\sum_{k=1}^{K}c_{k}\prod_{i=1}^{N}x_{i}^{a_{ik}}$
(2)

From these two building blocks, it is possible to construct the definition of
a GP in standard form boyd2007tutorial :

$\displaystyle\underset{\mathbf{x}}{\text{minimize}}$ $\displaystyle
p_{0}(\textbf{x})$ (3) subject to $\displaystyle
m_{i}(\textbf{x})=1,\;i=1,\ldots,N$ $\displaystyle p_{j}(\textbf{x})\leq
1,\;j=1,\ldots,M$

When constraints and objectives can be written in the form specified in
Equation 3 it is said that the problem is GP compatible.

###  2.2 Signomial Programming

Signomal Programs (SPs) are a logical extension of Geometric Programs that
allow the inclusion of negative leading constants and a broader set of
equality constraints. The key building blocks of the Signomial Programing are
signomials, which are the difference between two posynomials $p(\textbf{x})$
and $n(\textbf{x})$:

$s(\textbf{x})=p(\textbf{x})-n(\textbf{x})=\sum_{k=1}^{K}c_{k}\prod_{i=1}^{N}x_{i}^{a_{ik}}-\sum_{p=1}^{P}d_{p}\prod_{i=1}^{N}x_{i}^{g_{ik}}$
(4)

The posynomial $n(\textbf{x})$ is often referred to as a ‘neginomial’ because
it is made up of all of the terms with negative leading coefficients. From
this definition, the standard form for a Signomial Program can be constructed
kirschen2016signomial :

$\displaystyle\underset{\mathbf{x}}{\text{minimize}}$
$\displaystyle\frac{p_{0}(\textbf{x})}{n_{0}(\textbf{x})}$ (5) subject to
$\displaystyle s_{i}(\textbf{x})=0,\;i=1,\ldots,N$ $\displaystyle
s_{j}(\textbf{x})\leq 0,\;j=1,\ldots,M$

however, another useful form is kirschen2016signomial :

$\displaystyle\underset{\mathbf{x}}{\text{minimize}}$
$\displaystyle\frac{p_{0}(\textbf{x})}{n_{0}(\textbf{x})}$ (6) subject to
$\displaystyle\frac{p_{i}(\textbf{x})}{n_{i}(\textbf{x})}=1,\;i=1,\ldots,N$
$\displaystyle\frac{p_{j}(\textbf{x})}{n_{j}(\textbf{x})}\leq
1,\;j=1,\ldots,M$

In this alternative form, the neginomial is added to both sides, and then used
as a divisor to construct an expression either equal to or constrained by a
value of one.

SPs are not convex upon log-log transformation unlike their GP counterparts
and therefore must be solved using general non-linear solution methods.
However many signomial programs of interest still benefit from an underlying
structure which is well approximated by a log-log convex formulation, and as a
result can be efficiently solved by a series of GP approximations via the
Difference of Convex Algorithm (DCA). In this process the various neginomials
$n(\textbf{x})$ are replaced with local monomial approximations, yielding
substantial benefits over other non-linear solution methods (see
kirschen2018application  and york2018efficient  for discussion).

##  3 Difference of Convex Functions for Data Fitting

###  3.1 Difference of Convex (DC) Functions

Signomials are extremely versatile functions, far more versatile than
posynomials or monomials. But the construction of signomials is rather unique
since they are the difference between two log-log convex functions.

Modern study of Difference of Convex (DC) functions is widely attributed to
have started with a paper by Hartman et. al. hartman1959functions , which
states that a continuous function of bounded variation ($f(\mathbf{x})$) can
be written as the difference of two convex functions ($g(\mathbf{x})$ and
$h(\mathbf{x})$) hartman1959functions :

$f(\mathbf{x})=g(\mathbf{x})-h(\mathbf{x})$ (7)

which clearly mirrors signomial construction. Further extensions have opened
up a wide variety of well known properties that DC functions possess
bavcak2011difference ; rudin2012prediction .

The implication here is that a significant subset of continuous functions can
be decomposed into the difference of two convex functions, including non-
differentiable continuous functions. By extension, it should be possible to
fit a DC function to nearly any dataset that can be well approximated with a
continuous function.

Hoburg et. al. hoburg2016data  provided three ways of fitting functions to
datasets that were convex under the log-log transformation, but these methods
entirely ignore the additional modeling flexibility available in Signomial
Programming. So consider the use of a DC function under the log-log
transformation, specifically the difference between two of the functions
proposed by Hoburg et. al. hoburg2016data .

###  3.2 Function Definitions

####  3.2.1 Notation

Consider a data set sampled from a black box mapping from
$\mathds{R}^{N}\rightarrow\mathds{R}$. Consistent with the notation in Hoburg
hoburg2016data  let the vector $\mathbf{u}_{j}$ represent the independent
variables in $\mathds{R}^{N}$ for data point $j$ and the scalar $w_{j}$
represent the output in $\mathds{R}$ for data point $j$. The logspace
variables are then represented as $\mathbf{x}_{j}=\log\mathbf{u}_{j}$ and
$y_{j}=\log w_{j}$.

####  3.2.2 Difference of Max Affine (DMA) Functions

The first function proposed by Hoburg et. al. hoburg2016data  is the Max
Affine (MA) function:

$f_{\text{MA}}(\textbf{x})=\max_{k=1\ldots
K}\left[b_{k}+\textbf{a}_{k}^{\text{T}}\textbf{x}\right]$ (8)

This function class is known to create a convex epigraph. In fact, any convex
function can be reasonably approximated as a max affine function given a
sufficient number of affine functions, $K$.

Now consider the difference between two of these max affine functions
(Equation 8), which henceforth will be called the Difference of Max Affine
(DMA) function:

$f_{\text{DMA}}(\textbf{x})=\max_{k=1\ldots
K}\left[b_{k}+\textbf{a}_{k}^{\text{T}}\textbf{x}\right]-\max_{m=1\ldots
M}\left[h_{m}+\textbf{g}_{m}^{\text{T}}\textbf{x}\right]$ (9)

The subtracted term is represented by an entirely separable Max Affine
function which is defined by fitting parameters $M$, $h$, and $\mathbf{g}$.

While the Max Affine function has a realizable transformation back from
logspace, the Difference of Max Affine function has no such transformation due
to the inability to readily construct a meaningful epigraph or subgraph.
Despite this, the DMA function is quite rapid to fit, and could be of
application in other areas where a cheap surrogate is desired for non-convex
fitting. Here, the DMA function is a useful as an intellectual building block,
and as a seed for the function class that follows.

####  3.2.3 Soft Difference of Max Affine (SDMA) Functions

The second function proposed by Hoburg et. al. hoburg2016data  is the Soft-Max
Affine (SMA) function:

$f_{\text{SMA}}(\textbf{x})=\frac{1}{\alpha}\log\sum_{k=1}^{K}\exp\left(\alpha(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x})\right)$
(10)

The SMA function uses a global softening parameter ($\alpha$) to ‘smooth’ the
sharp corners of the Max Affine function and has the benefit of requiring far
fewer affine terms $K$ to capture smooth convex functions with reasonable
accuracy. However, the global softening parameter results in a poor
representation in regions where the curvature deviates substantially from the
global average.

Consider the following function which is the difference between two Soft-Max
Affine functions (Equation 10):

$f_{\text{SDMA}}(\textbf{x})=\frac{1}{\alpha}\log\sum_{k=1}^{K}\exp\left(\alpha(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x})\right)-\frac{1}{\beta}\log\sum_{m=1}^{M}\exp\left(\beta(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x})\right)$
(11)

which his henceforth referred to as the Soft Difference of Max Affine (SDMA)
function. As with SMA functions when compared to MA functions, the SDMA
function compared to the DMA function significantly decreases the number of
terms $K+M$ required to approximate smooth functions to a reasonable degree of
accuracy by introducing scalar softening parameters $\alpha$ and $\beta$.

Transforming the SDMA function back to the optimization variables
$\mathbf{u}_{j}$ and $w_{j}$ proceeds as follows:

$\displaystyle y$
$\displaystyle=\frac{1}{\alpha}\log\sum_{k=1}^{K}\exp\left(\alpha(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x})\right)-\frac{1}{\beta}\log\sum_{m=1}^{M}\exp\left(\beta(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x})\right)$
(12) $\displaystyle\exp(y)$
$\displaystyle=\exp\left(\frac{1}{\alpha}\log\sum_{k=1}^{K}\exp\left(\alpha(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x})\right)-\frac{1}{\beta}\log\sum_{m=1}^{M}\exp\left(\beta(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x})\right)\right)$
$\displaystyle w$
$\displaystyle=\frac{\exp\left(\frac{1}{\alpha}\log\sum_{k=1}^{K}\exp\left(\alpha(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x})\right)\right)}{\exp\left(\frac{1}{\beta}\log\sum_{m=1}^{M}\exp\left(\beta(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x})\right)\right)}$
$\displaystyle w$
$\displaystyle=\frac{\exp\left(\log\left(\sum_{k=1}^{K}\exp\left(\alpha(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x})\right)\right)^{\frac{1}{\alpha}}\right)}{\exp\left(\log\left(\sum_{m=1}^{M}\exp\left(\beta(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x})\right)\right)^{\frac{1}{\beta}}\right)}$

At this point it is obvious from the definition of a posynomial function that
the form will reduce to:

$w=\frac{\left(\sum_{k=1}^{K}e^{\alpha b_{k}}\prod_{i=1}^{N}u_{i}^{\alpha
a_{ik}}\right)^{1/\alpha}}{\left(\sum_{m=1}^{M}e^{\beta
h_{m}}\prod_{i=1}^{N}u_{i}^{\beta g_{im}}\right)^{1/\beta}}$ (13)

This is not compatible with the SP formulation, but consider the following
substitutions:

$\displaystyle p_{convex}$ $\displaystyle=\sum_{k=1}^{K}e^{\alpha
b_{k}}\prod_{i=1}^{N}u_{i}^{\alpha a_{ik}}$ (14) $\displaystyle p_{concave}$
$\displaystyle=\sum_{m=1}^{M}e^{\beta h_{m}}\prod_{i=1}^{N}u_{i}^{\beta
g_{im}}$ (15)

Which then reduces Equation 13 to:

$w=\frac{(p_{convex})^{1/\alpha}}{(p_{concave})^{1/\beta}}$ (16)

Thus, taking the three constraints (Equations 14, 15, 16) as a set does result
in an SP compatible scheme. This method of substitution is consistent with
other approaches to constructing GP and SP compatible constraints
boyd2007tutorial . The table below outlines the operators that should be used
in the new constraint set given the original mapping:

Table 1: Operators For a Given Initial Constraint | Constraint Operator
---|---
Original |  $=$ |  $\leq$ |  $\geq$
$w$ |  $=$ |  $\leq$ |  $\geq$
$p_{convex}$ |  $=$ |  $\geq$ |  $\leq$
$p_{concave}$ |  $=$ |  $\leq$ |  $\geq$

####  3.2.4 Consideration of Implicit Soft Difference of Max Affine (ISDMA)
Functions

Hoburg et. al. hoburg2016data  proposes a third function class, Implicit Soft
Max Affine (ISMA). Unlike MA and SMA, the ISMA function class is an implicit
function of $y$:

$\displaystyle\tilde{f}_{\text{ISMA}}(\mathbf{x},y)$
$\displaystyle=\log\sum_{k=1}^{K}\alpha_{k}\exp\left(\alpha_{k}(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x}-y)\right)$
(17)

Since the ISMA function class provided a fit with lower RMS error than SMA, it
is tempting to write an implicit soft difference of max affine function as:

$\displaystyle\tilde{f}_{\text{ISDMA}}(\mathbf{x},y)$
$\displaystyle=\log\sum_{k=1}^{K}\alpha_{k}\exp\left(\alpha_{k}(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x}-y_{convex})\right)$
(18)
$\displaystyle-\log\sum_{m=1}^{M}\beta_{m}\exp\left(\beta_{m}(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x}-y_{concave})\right)$

Fitting one of these functions to data would require that the data first be
decomposed into convex ($y_{convex}$) and concave ($y_{concave}$), which has
no obvious solution a priori. Fortunately, in practice the SDMA function is
highly versatile and captures regions with varying curvature for even low fit
orders due to the DC construction, somewhat negating the desire for ISDMA
functions in the first place. However, non-differentiable corners and cusps
would benefit from this type of function if the issues could be successfully
resolved.

##  4 How Functions are Fit

The following process is used for fitting functions to the data points
($\mathbf{u}_{j}$,$w_{j}$):

  1. 1.

Apply the log transformation $(\mathbf{x}_{j},y_{j})=(\log\mathbf{u}_{j},\log
w_{j})$

  2. 2.

Fit a function $f$ to the transformed data such that $y_{j}\approx
f(\mathbf{x}_{j})$

  3. 3.

Relax the equality $y_{j}=f(\mathbf{x}_{j})$ as appropriate to construct the
desired constraint

This process is nearly identical to that proposed by Hoburg et. al.
hoburg2016data , but two differences exist. First, the work here does not
require convex functions in Step 2. Second, while Hoburg et. al.
hoburg2016data  could readily relax each equality constraint to construct a
convex epigraph in Step 3, there is no similar approach when considering
functions that are generally not convex and so selecting the appropriate
constraint operator ($=$,$\leq$, or $\geq$) will depend on the constraint
being considered.

For a set of $m$ data points, the fitting problem can be written as an
unconstrained least squares optimization problem:

$\underset{\gamma}{\text{minimize}}\hskip
4.30554pt\sum_{i=1}^{m}\left(f(\mathbf{x}_{i};\mathbf{\gamma})-y_{i}\right)^{2}$
(19)

where the fitting parameters are stacked in the vector $\gamma$.

Based on the work of Hoburg hoburg2016data , all of the functions presented
here were fit using a Levenberg-Marquardt (LM) algorithm. The LM algorithm
computes a step size at each iteration, but requires a Jacobian to be computed
at each step. Derivatives are therefore presented for each function class with
respect to the fitting parameters in the vector $\gamma$ in the two sections
that follow.

For each SDMA function fit, a DMA function is first fit in order to provide an
initial guess for the SDMA fitting algorithm. The ability to quickly achieve a
DMA initial guess is critical to the success of SDMA functions, as starting
from random positions rarely yields a good result.

In addition, the gradient based nature of the LM algorithm meant that the
fitting process required a number of random restarts to be performed from
varying initial guesses. In general 30-100 random restarts were performed for
the work presented here, with the specific number varying with the individual
fitting problem being considered.

Non-gradient based methods such as a Genetic Algorithm or Particle Swarm
Optimization were considered for fitting, but there was no evidence that this
approach would be superior to the LM algorithm established in the literature.
A Genetic Algorithm was actually tested, but had worse performance than the LM
method even in simple 2D cases.

###  4.1 Derivatives for the DMA Function

$\frac{\partial f_{\text{DMA}}}{\partial b_{i}}=\left\\{\begin{tabular}[]{l}1,
\text{if} $i=\text{argmax}_{k}\hskip
4.30554ptb_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x}$\\\ 0,
otherwise\end{tabular}\right.$ (20) $\hskip 8.61108pt\frac{\partial
f_{\text{DMA}}}{\partial\mathbf{a}_{i}}=\left\\{\begin{tabular}[]{l}$\mathbf{x}^{\text{T}}$,
\text{if} $i=\text{argmax}_{k}\hskip
4.30554ptb_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x}$\\\ 0,
otherwise\end{tabular}\right.$ (21) $\frac{\partial f_{\text{DMA}}}{\partial
h_{i}}=\left\\{\begin{tabular}[]{l}-1, \text{if} $i=\text{argmax}_{k}\hskip
4.30554ptb_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x}$\\\ 0,
otherwise\end{tabular}\right.$ (22) $\hskip 8.61108pt\frac{\partial
f_{\text{DMA}}}{\partial\mathbf{g}_{i}}=\left\\{\begin{tabular}[]{l}$-\mathbf{x}^{\text{T}}$,
\text{if} $i=\text{argmax}_{k}\hskip
4.30554ptb_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x}$\\\ 0,
otherwise\end{tabular}\right.$ (23)

###  4.2 Derivatives for the SDMA Function

$\displaystyle\frac{\partial f_{\text{SDMA}}}{\partial b_{i}}$
$\displaystyle=\frac{\exp({\alpha(b_{i}+\mathbf{a}_{i}^{\text{T}}\mathbf{x}))}}{\sum_{k=1}^{K}\exp(\alpha(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x}))}$
$\displaystyle\frac{\partial f_{\text{SDMA}}}{\partial\mathbf{a}_{i}}$
$\displaystyle=\frac{\mathbf{x}^{\text{T}}\cdot\exp(\alpha(b_{i}+\mathbf{a}_{i}^{\text{T}}\mathbf{x}))}{\sum_{k=1}^{K}\exp(\alpha(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x}))}$
$\displaystyle\frac{\partial f_{\text{SDMA}}}{\partial\alpha}$
$\displaystyle=\frac{1}{\alpha}\frac{\sum_{k=1}^{K}(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x})\exp(\alpha(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x}))}{\sum_{k=1}^{K}\exp(\alpha(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x}))}$
$\displaystyle-\frac{1}{\alpha^{2}}\log\sum_{k=1}^{K}\exp\left(\alpha(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x})\right)$
$\displaystyle\frac{\partial f_{\text{SDMA}}}{\partial(\log\alpha)}$
$\displaystyle=\frac{\sum_{k=1}^{K}(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x})\exp(\alpha(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x}))}{\sum_{k=1}^{K}\exp(\alpha(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x}))}$
$\displaystyle-\frac{1}{\alpha}\log\sum_{k=1}^{K}\exp\left(\alpha(b_{k}+\mathbf{a}_{k}^{\text{T}}\mathbf{x})\right)$
$\displaystyle\frac{\partial f_{\text{SDMA}}}{\partial h_{i}}$
$\displaystyle=\frac{-\exp({\beta(h_{i}+\mathbf{g}_{i}^{\text{T}}\mathbf{x}))}}{\sum_{m=1}^{M}\exp(\beta(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x}))}$
$\displaystyle\frac{\partial f_{\text{SDMA}}}{\partial\mathbf{g}_{i}}$
$\displaystyle=\frac{-\mathbf{x}^{\text{T}}\cdot\exp(\beta(h_{i}+\mathbf{g}_{i}^{\text{T}}\mathbf{x}))}{\sum_{m=1}^{M}\exp(\beta(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x}))}$
(24) $\displaystyle\frac{\partial f_{\text{SDMA}}}{\partial\beta}$
$\displaystyle=-\frac{1}{\beta}\frac{\sum_{m=1}^{M}(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x})\exp(\beta(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x}))}{\sum_{m=1}^{M}\exp(\beta(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x}))}$
$\displaystyle~{}~{}~{}+\frac{1}{\beta^{2}}\log\sum_{m=1}^{M}\exp\left(\beta(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x})\right)$
$\displaystyle\frac{\partial f_{\text{SDMA}}}{\partial(\log\beta)}$
$\displaystyle=-\frac{\sum_{m=1}^{M}(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x})\exp(\beta(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x}))}{\sum_{m=1}^{M}\exp(\beta(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x}))}$
$\displaystyle~{}~{}~{}+\frac{1}{\beta}\log\sum_{m=1}^{M}\exp\left(\beta(h_{m}+\mathbf{g}_{m}^{\text{T}}\mathbf{x})\right)$

##  5 Demonstrations of the Fitting Models

###  5.1 A 2D Fitting Problem

Consider the 2D function, which uses the log transformed variables
$\mathbf{x}$ and $y$:

$y=\max\left[-6x-6,x^{4}-3x^{2}\right]$ (25)

This function was selected to test the SDMA function at a non-differentiable
point and over variations in curvature. A set of 101 data points were sampled
on even intervals over the function from $x=[-2,2]$, and 5th order MA, SMA,
ISMA, and SDMA functions were fit to this data:

Figure 1: Fifth order fits to the simple 2D function. RMS Error is as follows:
MA: 44.6%, SMA: 44.6%, ISMA: 44.8%, SDMA: 0.149%

The convex fitting methods all converge to nearly identical representations,
none of which capture the significant non-convexity to any degree. In
contrast, the SDMA function captures nearly all of the non-convexity in the
function, to an RMS error of $1.5\times 10^{-3}$. The key takeaway here is
that unlike SMA functions, which can only capture a single curvature due to
the single parameter $\alpha$, the SDMA function can capture complex, multi-
radius curvature as a direct result of the DC construction.

Increasing fit order also improves the RMS error as expected:

Figure 2: Variation of RMS error with increasing fit order

###  5.2 A 3D Fitting Problem

After proving on a 2D case, the next logical step was to consider a 3D
dataset. Similar to the 2D test case, it was desirable to have a non-
differentiable region and complex 3D curvature. Consider this test function,
the eigenfunction of the wave equation:

Figure 3: The eigenfunction of the wave equation

This function has become well known as the Matlab logo.

An SDMA fit yields a reasonably accurate surrogate (consider here 9th order):

(a) The fitted 9th order function

(b) Error compared to the true function

Figure 4: The fitted 9th order function and error. Note the increased value at
the top of the color axis compared to 8th order, indicating a larger maximum
error

As might be expected, the fitting scheme struggles along the non-
differentiable L-shaped curve, and at the point of the L specifically. Error
near the point (0,1) is due to an inflection in the data where curvature
changes from concave to convex in a very small region, and this is difficult
to capture in the fit.

And as before, plotting RMS error as a function of increasing fit order shows
significant improvement for higher order fits:

Figure 5: Variation of RMS error with increasing fit order

###  5.3 Fitting XFOIL Performance Data of the NACA 24xx Family of Airfoils

Given the success of the two test cases, the final step was to validate on a
more representative engineering example. The Hoburg et. al. hoburg2016data
work was able to fit performance data for NACA 24xx airfoils generated from
XFOIL drela1989xfoil , but only by considering curves of lift coefficient vs.
drag coefficient. In many cases, it is more useful to have two separate curves
of lift coefficient vs. angle of attack and drag coefficient vs. angle of
attack, but the CL vs. $\alpha$ curve is not compatible with log-log convex
fitting techniques.

Consider the problem of fitting the following function:

$C_{L}=f(\alpha,Re,\tau)$ (26)

where $C_{L}$ is the airfoil lift coefficient, $\alpha$ is the airfoil angle
of attack, $Re$ is the Reynolds number, and $\tau$ is the airfoil thickness
(ex, a $\tau=0.12$ would be a NACA 2412 airfoil).

XFOIL was used to generate a grid of data from $\alpha=[1,23]$,
$Re=[10^{6},10^{7}]$, and $\tau=[.09,.21]$. This data was then fit with an
SDMA function of fit orders varying from 3 to 10. Plotting below are the
results of a 5th order fit:

(a) Reynolds Number of $10^{6}$

(b) Reynolds Number of $10^{6.5}$

(c) Reynolds Number of $10^{7}$

Figure 6: Curves of Lift Coefficient vs. Angle of Attack for the NACA 24xx
family of airfoils

As before, plotting the RMS error vs. increasing fit order shows improvement
with an increased number of fit terms:

Figure 7: Variation of RMS error with increasing fit order

##  6 Conclusions

This work serves as a successful development and validation of a Signomial
Programming compatible method for fitting black box data. The Soft Difference
of Max Affine (SDMA) function is a crucial link in the chain from Geometric
Programming, where low fidelity analysis methods must be used, to Sequential
Quadratic Programming or the more recently proposed Logspace Sequential
Quadratic Programming karcher2021logspace , which typically impose no bounds
on the type of analysis that can be utilized.

###### Acknowledgements.

This material is based on research sponsored by the U.S. Air Force under
agreement number FA8650-20-2-2002. The U.S. Government is authorized to
reproduce and distribute reprints for Governmental purposes notwithstanding
any copyright notation thereon. The views and conclusions contained herein are
those of the authors and should not be interpreted as necessarily representing
the official policies or endorsements, either expressed or implied, of the
U.S. Air Force or the U.S. Government. The author would like to thank Bob
Haimes, Woody Hoburg, and Berk Ozturk for their input into the technical
matter, along with the EnCAPS Technical Monitor Ryan Durscher.

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# Integral Zariski dense surface groups in $\SL(n,\R)$

Michael Zshornack  Department of Mathematics, University of California Santa
Barbara, Santa Barbara, CA 93106, USA<EMAIL_ADDRESS>https://sites.google.com/view/michael-zshornack

###### Abstract.

Given a number field $K$, we show that certain $K$-integral representations of
closed surface groups can be deformed to being Zariski dense while preserving
many useful properties of the original representation. This generalizes a
method due to Long and Thistlethwaite who used it to show that thin surface
groups in $\operatorname{SL}(2k+1,\mathbf{Z})$ exist for all $k$.

###### 1991 Mathematics Subject Classification:

57K20

##  1\. Introduction

Thin groups are a class of groups which have a rich number of arithmetic
properties that have been of interest in recent years. Given a finitely
generated subgroup $\Gamma<\operatorname{GL}(n,\mathbf{Z})$, we let $G$ denote
its Zariski closure: $G:=\operatorname{Zcl}(\Gamma)$. We say that the group
$\Gamma$ is thin if it is of infinite index in $G(\mathbf{Z})$ (see [Sar14]).
Tools used to study these groups, such as Super Approximation, expander
families and the affine sieve, have been rich areas of research that have seen
a lot of progress in the last 10–15 years (see [BG08] and [BGS10]).

One large class of examples of these come from free subgroups which are thin
in $\operatorname{SL}(n,\R)$. Finding such subgroups is fairly well understood
(see, for example, [FR16]). Much less well understood is in subgroups which
are thin in $\operatorname{SL}(n,\R)$ and do not decompose as a free product
of two smaller subgroups. In this paper, we seek to further understand ways in
which one might construct freely indecomposable thin subgroups of
$\operatorname{SL}(n,\R)$ through methods of low-dimensional topology.

We let $S=S_{g}$ denote a connected, closed, orientable surface of genus
$g\geq 2$. Our goal will be to use techniques available to us from higher
Teichmüller theory to construct discrete and faithful representations of
$\pi_{1}(S)$ which are Zariski dense in $\operatorname{SL}(n,\R)$.

The starting point of this construction is to look at points on the Hitchin
component of $S$, which we denote by $\operatorname{Hit}_{n}(S)$. These are
components of the $\operatorname{PSL}(n,\R)$-character variety of $S$,
$\mathcal{R}_{n}(S)$, which contain the image of Teichmüller space under the
map $\mathcal{T}(S)\to\mathcal{R}_{n}(S)$ given by composing a Fuchsian
representation $\rho_{0}:\pi_{1}(S)\to\operatorname{PSL}(2,\R)$ with the
irreducible representation
$\tau_{n}:\operatorname{PSL}(2,\R)\to\operatorname{PSL}(n,\R)$. These
components were studied by Hitchin in [Hit92] and later by Labourie in [Lab06]
where the authors showed that the representations on these components had many
of the same properties which were classical facts about representations in
Teichmüller space (which we briefly summarize in §2). In particular, these
properties of the Hitchin component ensure that the representations we
construct will be discrete, faithful and can be lifted to
$\operatorname{SL}(n,\R)$, thus providing a method for constructing subgroups
of the form we seek.

This method of producing Zariski dense surface subgroups of
$\operatorname{SL}(n,\R)$ originates in [LT22] where the authors proved the
existence of surface subgroups of $\operatorname{SL}(2k+1,\mathbf{Z})$ for
arbitrary $k\geq 1$ whose image was Zariski dense in
$\operatorname{SL}(2k+1,\R)$. The starting point for their construction is the
representation coming from the hyperbolic structure on the orbifold with
signature $S^{2}(3,4,4)$ (which is unique up to conjugacy due to these
orbifolds being rigid). Direct calculation shows that the image of the
representation coming from the holonomy of this hyperbolic structure can be
conjugated into $\operatorname{SL}(n,\mathbf{Z})$ after composing with
$\tau_{n}$ for $n$ odd. The authors took this representation and utilized a
bending construction of Thurston to deform it so that, after passing to a
surface subgroup of finite index, they were left with a representation of a
surface group into $\operatorname{SL}(2k+1,\mathbf{Z})$ with Zariski dense
image. Various tools in the theory of Hitchin representations and the theory
of algebraic groups were needed in their proof to ensure this bending
construction could be done while preserving certain properties of the original
representation, but this result provided the first examples of freely
indecomposable isomorphism classes of thin subgroups of
$\operatorname{SL}(n,\R)$ for infinitely many $n$.

In this paper, we seek to generalize the methods used in [LT22] to show this
process of bending an integral representation to being Zariski dense can also
be done in a much more general context to produce more examples of Zariski
dense subgroups of $\operatorname{SL}(n,\R)$. For any subring $R\subseteq\R$,
we will call a representation $\rho:\pi_{1}(S)\to\operatorname{SL}(n,\R)$ an
$R$-point of the character variety if it can be conjugated in
$\operatorname{SL}(n,\R)$ to have image in $\operatorname{SL}(n,R)$.
Throughout the rest of this paper, we will let $K/\mathbf{Q}$ be a number
field with ring of integers $\mathcal{O}_{K}$. We assume that $K\subset\R$ (so
in particular, $K$ is not totally imaginary) and that $K$ has class number
one. The class number one hypothesis is mainly for technical reasons and can
perhaps be removed in future work, but our main result is the following.

######  Theorem 1.1.

For any $n\geq 2$, if $\operatorname{Hit}_{n}(S)$ contains an
$\mathcal{O}_{K}$ point, then there is a finite sheeted cover
$\widetilde{S}\to S$ so that $\operatorname{Hit}_{n}(\widetilde{S})$ contains
an $\mathcal{O}_{K}$-point whose image is Zariski dense in
$\operatorname{SL}(n,\R)$.

######  Remark.

When $K\neq\mathbf{Q}$, we prove an even stronger result (Theorem 4.3) that
circumvents the need to pass to a finite sheeted cover.

At its core, the proof of Theorem 1.1 is a construction of a similar nature to
the one presented in [LT22] with some key modifications made at various steps
to handle the slightly more general setup. In addition to handling the even
$n$ case, this theorem utilizes a slightly different version of the bending
construction more suited to the topology of $S$. These modifications also lend
themselves to some other simplifications in the proof which we will discuss
later.

When we specialize Theorem 1.1 to the case where $K=\mathbf{Q}$, we will be
left with a representation of $\pi_{1}(S)$ into
$\operatorname{SL}(n,\mathbf{Z})$ with Zariski dense image in
$\operatorname{SL}(n,\R)$. The image also is of infinite index in
$\operatorname{SL}(n,\mathbf{Z})$ due to results of Margulis. Namely, since
surface groups certainly surject onto $\mathbf{Z}$, the image will
automatically be of infinite index in $\operatorname{SL}(n,\mathbf{Z})$ hence
thin (see [Mar91]). This representation will be on the Hitchin component,
hence faithful, and so the image will define a freely indecomposable thin
sugroup of the form we wanted. Therefore, Theorem 1.1 leads to the following
useful corollary on the existence of freely indecomposable thin subgroups of
$\operatorname{SL}(n,\R)$.

######  Corollary 1.1.1.

If $\operatorname{Hit}_{n}(S)$ contains a $\mathbf{Z}$-point, then
$\operatorname{SL}(n,\mathbf{Z})$ contains a thin surface subgroup.

Thus the problem of finding non-free thin subgroups of
$\operatorname{SL}(n,\R)$ can be reduced to finding integral representations
of surface groups on Hitchin components. Of course, finding such
representations still remains a difficult problem and at the time of writing,
examples of these are only known in the case where $n=2k+1$ is odd or $n=4$.

#### Acknowledgments

The author would like to thank Darren Long for the guidance provided
throughout the writing of this paper. In particular, his numerous helpful
discussions held with the author and edits of early drafts were instrumental
in the completion of this work.

##  2\. Properties of Hitchin representations

The input to the construction used to prove Theorem 1.1 is a representation
$\rho:\pi_{1}(S)\to\operatorname{PSL}(n,\mathcal{O}_{K})$ lying on the Hitchin
component of $\mathcal{R}_{n}(S)$. We start with representations on such
components because they possess many useful properties akin to the
representations in $\mathcal{T}(S)$.

######  Theorem 2.1 (Labourie, [Lab06]).

If $\rho:\pi_{1}(S)\to\operatorname{PSL}(n,\R)$ belongs to the Hitchin
component, it is discrete, faithful, purely loxodromic and strongly
irreducible.

In this context, $\rho$ is purely loxodromic if and only if for every non-
identity element $\gamma\in\pi_{1}(S)$, $\rho(\gamma)$ is diagonalizable with
real eigenvalues, all with distinct absolute values. We also say $\rho$ is
strongly irreducible if the restriction of $\rho$ to any finite index subgroup
of $\pi_{1}(S)$ is irreducible.

Alessandrini, Lee and Schaffhauser have also generalized the conclusions of
Theorem 2.1 to also hold in the case where $X$ is a compact connected
$2$-dimensional orbifold with boundary and $\chi(X)<0$. Their result (Theorem
2.28 in [ALS20]) provides many of the same properties for non-closed surfaces
which will be necessary for the bending construction of §3.

The main utility of this theorem is that the bending process we use to deform
$\rho$ preserves the path component $\rho$ lies on in $\mathcal{R}_{n}(S)$.
Thus as $\rho$ originally lied on the Hitchin component, then so will the bent
representation, hence it will still be discrete, faithful and purely
loxodromic.

We may also circumvent the (mild) annoyance of having to work with projective
matrices due to the fact that representations in $\mathcal{T}(S)$ can be
lifted to $\operatorname{SL}(2,\R)$ and the fact that the Hitchin component is
connected. The following fact is known, but we record a proof here as well.

######  Lemma 2.2.

Let $\rho:\pi_{1}(S)\to\operatorname{PSL}(n,\R)$ be a representation on the
Hitchin component. Then $\rho$ admits a lift
$\widetilde{\rho}:\pi_{1}(S)\to\operatorname{SL}(n,\R)$. Moreover, for any
nontrivial $\gamma\in\pi_{1}(S)$, the lift $\widetilde{\rho}$ can be chosen so
that $\widetilde{\rho}(\gamma)$ has distinct positive real eigenvalues.

###### Proof.

We start with the following presentation of $\pi_{1}(S)$:

$\pi_{1}(S)=\langle
a_{1},b_{1},\ldots,a_{g},b_{g}:[a_{1},b_{1}]\ldots[a_{g},b_{g}]=1\rangle.$

Following [Gol88], fix lifts $\widetilde{\rho}(a_{i})$ and
$\widetilde{\rho}(b_{i})\in\operatorname{SL}(n,\R)$ for each $i=1,\ldots,g$.
The question of whether or not these choices define a representation
$\widetilde{\rho}:\pi_{1}(S)\to\operatorname{SL}(n,\R)$ depends on the value
of the element

$[\widetilde{\rho}(a_{1}),\widetilde{\rho}(b_{1})]\ldots[\widetilde{\rho}(a_{g}),\widetilde{\rho}(b_{g})]\in\left\\{\pm
I_{n}\right\\}.$

Thus, the obstruction to lifting
$\rho\in\operatorname{Hom}(\pi_{1}(S),\operatorname{PSL}(n,\R))$ to
$\widetilde{\rho}\in\operatorname{Hom}(\pi_{1}(S),\operatorname{SL}(n,\R))$
defines a map

$o:\operatorname{Hom}(\pi_{1}(S),\operatorname{PSL}(n,\R))\to\\{\pm I_{n}\\}.$

This map must be locally constant as small deformations of $\rho$ don’t change
the value of the above produce of commutators of lifts and $\rho$ is a Hitchin
representation, it lies on the same component as the representations coming
from Teichmüller space. Therefore, it suffices to show that a discrete and
faithful representation $\rho:\pi_{1}(S)\to\operatorname{PSL}(2,\R)$ lifts to
$\operatorname{SL}(2,\R)$.

In this case, any representation $\rho:\pi_{1}(S)\to\operatorname{PSL}(2,\R)$
defines a circle bundle over $S$ given by $(\widetilde{S}\times
S^{1})/\rho(\pi_{1}(S))$, where the action on $S^{1}$ is given by the natural
action $\rho$ induces on the circle at infinity of $\mathbf{H}^{2}$. In this
case, the representation $\rho:\pi_{1}(S)\to\operatorname{PSL}(2,\R)$ lifts to
$\operatorname{SL}(2,\R)$ if and only if the Euler number of this circle
bundle is divisible by $2$. When $\rho$ is a discrete and faithful
representation, the circle bundle $\rho$ defines is topologically equivalent
to the unit tangent bundle of $S$, which has Euler number $\pm(2g-2)$, and so
$\rho$ will indeed lift to some
$\widetilde{\rho}:\pi_{1}(S)\to\operatorname{SL}(2,\R)$.

We can also guarantee that the lifts can be chosen so that any nontrivial
element has distinct positive real eigenvalues as follows. Fix some nontrivial
$\gamma\in\pi_{1}(S)$. Since the eigenvalues of $\rho(\gamma)$ in
$\operatorname{PSL}(n,\R)$ are distinct, real and non-zero by 2.1, the set

$\left\\{\rho:\pi_{1}(S)\to\operatorname{PSL}(n,\R):\begin{tabular}[]{c}all
eigenvalues of a lift of $\rho(\gamma)$\\\ to $\operatorname{SL}(n,\R)$ have
the same sign\end{tabular}\right\\}$

is both open and closed in the Hitchin component. But then again, when $\rho$
is Fuchsian, we can write $\rho=\tau_{n}\rho_{0}$ for
$\rho_{0}:\pi_{1}(S)\to\operatorname{PSL}(2,\R)$ in $\mathcal{T}(S)$. Here, if
$\rho_{0}(\gamma)$ has eigenvalues $\lambda$ and $\lambda^{-1}$,
$\rho(\gamma)$ will have eigenvalues $\lambda^{n-2k+1}$ for $1\leq k\leq n$.
Thus for the lifts of Fuchsian representations, their eigenvalues have all the
same sign. So the above set is open and closed and nonempty, so must be all of
the Hitchin component. Thus, for any given $\gamma\in\pi_{1}(S)$, we can
always find a lift $\widetilde{\rho}:\pi_{1}(S)\to\operatorname{SL}(n,\R)$ so
that the eigenvalues of $\widetilde{\rho}(\gamma)$ have the same sign. If all
the eigenvalues happened to be negative, then we must be in the case where $n$
is even, in which case $-\widetilde{\rho}$ is also a lift of $\rho$ into
$\operatorname{SL}(n,\R)$ with the eigenvalues of $-\widetilde{\rho}(\gamma)$
all distinct and positive. ∎

Therefore, for the rest of this paper, we may assume our starting
representation on the Hitchin component takes the form
$\rho:\pi_{1}(S)\to\operatorname{SL}(n,\mathcal{O}_{K})$. For convenience as
well, whenever we fix a nontrivial $\gamma\in\pi_{1}(S)$, we will assume we
have chosen the lift of the representation so that $\rho(\gamma)$ then also
has distinct, positive real eigenvalues. Aside from just notational
convenience, these lifted representations will still be discrete and faithful
due to being on the Hitchin component and so their images will define non-free
subgroups of $\operatorname{SL}(n,\mathcal{O}_{K})$ of the form we wish to
study.

From this point, the goal will be to take the representation $\rho$ and
perform a series of deformations so that, after possibly passing to a finite
sheeted cover, we are left with a representation of a surface group into
$\operatorname{SL}(n,\mathcal{O}_{K})$ which is Zariski dense in
$\operatorname{SL}(n,\R)$. The following theorem will be crucial to
establishing this result.

######  Theorem 2.3 (Guichard, [Gui]).

Let $\rho:\pi_{1}(S)\to\operatorname{SL}(n,\R)$ be a lift of a representation
on the Hitchin component and let $G$ be the Zariski closure of
$\rho(\pi_{1}(S))$. Then $G$ is conjugate to one of the following groups:

  1. (i)

$\tau_{n}(\operatorname{SL}(2,\R))$.

  2. (ii)

$\operatorname{Sp}(2k,\R)$ if $n=2k$ is even.

  3. (iii)

$\operatorname{SO}(k+1,k)$ if $n=2k+1$ is odd.

  4. (iv)

The image of the $7$-dimensional fundamental representation of the short root
of $G_{2}$.

  5. (v)

$\operatorname{SL}(n,\R)$.

######  Remark.

An alternate proof of this same classification is also given in [Sam20].

We will assume $n\neq 7$ for simplicity so that the possible Zariski closures
will only be conjugate to options (i)–(iii) or (v) in the above list. From
here, we will show that if the Zariski closure of $\rho(\pi_{1}(S))$ is one of
(i)–(iii) in the above list, then there is a deformation of $\rho$ with the
properties that it is: $K$-integral, on the Hitchin component and has a
strictly larger Zariski closure. By performing this deformation multiple times
if necessary, we will be able to guarantee that the Zariski closure at the
final stage will be all of $\operatorname{SL}(n,\R)$ hence define a Zariski
dense subgroup. Guichard’s classification of possible Zariski closures is also
used in this manner in [LT22], but also relies on a generalization of the
result to representations of orbifold fundamental groups on Hitchin components
found in [ALS20].

##  3\. Bending around nonseparating curves

We now describe the bending construction which will be used to deform the
representations in order to successively enlarge the Zariski closures of the
image. For our surface $S=S_{g}$, we fix a nonseparating simple closed curve
$\gamma$. We let $S^{\prime}$ denote the compact surface one gets by cutting
$S$ along $\gamma$ and denote by $\gamma_{1}$ and $\gamma_{2}$ the two
boundary components of $S^{\prime}$. Up to some homeomorphism of $S$, we can
suppose our setup is of the form in Figure 1.

$\dotsc$$\gamma$$\dotsc$$\gamma_{1}$$\gamma_{2}$$S$$S^{\prime}$ Figure 1. The
setup for the bending construction.

In $S$, the curves $\gamma$, $\gamma_{1}$ and $\gamma_{2}$ will all be freely
homotopic, and by an abuse of notation, we will write $\gamma_{1}$ and
$\gamma_{2}$ to denote the elements of $\pi_{1}(S^{\prime})$ and identify
$\gamma$ with $\gamma_{1}$ in $\pi_{1}(S)$. In this setup, it is known that
$\pi_{1}(S)$ can be written as the following HNN extension:

$\pi_{1}(S)=\pi_{1}(S^{\prime})*_{\alpha}:=\langle\pi_{1}(S^{\prime}),\alpha\,|\,\alpha\gamma_{2}\alpha^{-1}=\gamma_{1}\rangle.$

Given any representation $\rho:\pi_{1}(S)\to\operatorname{SL}(n,\R)$ and any
matrix $A\in\operatorname{SL}(n,\R)$ in the centralizer of $\rho(\gamma)$, we
can define a new bent representation
$\rho^{A}:\pi_{1}(S)\to\operatorname{SL}(n,\R)$ by setting:

$\rho^{A}|_{\pi_{1}(S^{\prime})}:=\rho|_{\pi_{1}(S^{\prime})}\quad\textrm{and}\quad\rho^{A}(\alpha):=A\rho(\alpha).$

With extra care for the choice of centralizing $A$, this bending procedure
will also preserve a number of useful properties of the original
representation.

Firstly, if $\rho$ has image in $\operatorname{SL}(n,\mathcal{O}_{K})$ and we
choose $A$ in the $\mathcal{O}_{K}$-centralizer of $\rho(\gamma)$, then it’s
clear from the formula of the bend that $\rho^{A}$ still has image inside of
$\operatorname{SL}(n,\mathcal{O}_{K})$. So with additional care, the bending
construction can be done in such a way as to preserve $K$-integrality of the
original representation.

Secondly, if we take care to choose $A$ so that $A=\exp(X)$ for some
diagonalizable $X\in\mathfrak{sl}(n,\R)$, then there is a one-parameter family
of matrices given by

$A_{t}:=\exp(tX)\in\operatorname{SL}(n,\R),$

for $t\in\R$. For this one-parameter family, we have that $A_{0}=I_{n}$,
$A_{1}=A$. Moreover, by diagonalizability, for any $t\in\R$ $A_{t}$ still
centralizes $\rho(\gamma)$. Note that we are using the fact that
$\rho(\gamma)$ is also diagonalizable as $\rho$ lies on the Hitchin component
to guarantee the matrices $A_{t}$ all still centralize $\rho(\gamma)$. But in
particular, this one-parameter family of matrices $A_{t}$ defines a path of
representations $\rho^{A_{t}}:\pi_{1}(S)\to\operatorname{SL}(n,\R)$ from
$\rho^{A_{0}}=\rho$ to $\rho^{A_{1}}=\rho^{A}$. Thus, $\rho^{A}$ lies on the
same path component of $\mathcal{R}_{n}(S)$ as $\rho$. In particular, as
$\rho$ is a Hitchin representation, so is $\rho^{A}$.

So in short, many of the desired properties of the original representation can
be preserved by careful choice of the bending matrix $A$, used to deform
$\rho$. However, bending construction in this context is useful not just for
its ability to preserve “nice” properties of our representations, but to
“improve” them in the sense of producing representations with larger Zariski
closures. What ultimately enables this is the following.

######  Lemma 3.1.

Suppose $\rho(\pi_{1}(S))$ has Zariski closure contained in
$G\neq\operatorname{SL}(n,\R)$. If in addition to the above properties, we
choose $A\notin G$, then the Zariski closure of $\rho^{A}(\pi_{1}(S))$ is
strictly larger than $G$.

###### Proof.

The Zariski closure of $\rho^{A}(\pi_{1}(S))$ must still be among the ones of
Theorem 2.3 as $\rho^{A}$ is still on the Hitchin component of $S$. A priori,
it may be unclear why the image of $\rho^{A}$ might not be some nontrivial
conjugate of $G$ or a proper subgroup of it, but we can rule each of these
possibilities out as well, thus guaranteeing that the Zariski closure must
indeed get larger.

As $G\neq\operatorname{SL}(n,\R)$, consulting the list of Theorem 2.3, we see
that either $G$ is conjugate to either $\tau_{n}(\operatorname{SL}(2,\R))$,
$\operatorname{Sp}(2k,\R)$ or $\operatorname{SO}(k+1,k)$. In the first case,
$\rho^{A}(\pi_{1}(S))$ can’t have a smaller Zariski closure as no proper
subgroup of $\tau_{n}(\operatorname{SL}(2,\R))$ is a possible Zariski closure
for a Hitchin representation. In the latter two cases, it follows that there
is some alternating or symmetric bilinear form, $J$, which is preserved by the
image of $\rho$. Basic calculations verify that image of
$\operatorname{SL}(2,\R)$ under the irreducible representation $\tau_{n}$ is
contained in $\operatorname{Sp}(2k,\R)$ in the $n$ even case and
$\operatorname{SO}(k+1,k)$ in the $n$ odd case. Therefore, if
$\rho^{A}(\pi_{1}(S))$ had Zariski closure smaller than $G$, then the it must
be $\tau_{n}(\operatorname{SL}(2,\R))$ in which case, the same form $J$ is
still preserved. But this cannot be as $A$ won’t preserve $J$ by virtue of not
being in $G$.

All that remains is to rule out why the Zariski closure of
$\rho^{A}(\pi_{1}(S))$ can’t be some nontrivial conjugate of $G$. If this were
the case, then there would be another non-degenerate (symmetric or
alternating) bilinear form $J^{\prime}$ which is preserved by the image of
$\rho^{A}(\pi_{1}(S))$. But now, observe that the representations $\rho$ and
$\rho^{A}$ are identical on the compact subsurface $S^{\prime}\subset S$. In
particular, the image of the representation
$\rho|_{\pi_{1}(S^{\prime})}=\rho^{A}|_{\pi_{1}(S^{\prime})}$ preserves both
the forms $J$ and $J^{\prime}$.

As both $J$ and $J^{\prime}$ are preserved,
$J^{-1}J^{\prime}\rho(\beta)=\rho(\beta)J^{-1}J^{\prime}$ for all
$\beta\in\pi_{1}(S^{\prime})$. The subgroup $\pi_{1}(S^{\prime})$ has Zariski
closure at least containing an (absolutely irreducible) principal
$\operatorname{SL}(2,\R)$. $J^{-1}J^{\prime}$ centralizes all of this
principal $\operatorname{SL}(2,\R)$, hence by Schur’s lemma, is a homothety.
In particular, $J$ and $J^{\prime}$ are scalar multiples of one another and
define the same orthogonal or symplectic groups.

Thus the Zariski closure of $\rho^{A}(\pi_{1}(S))$ cannot be a subgroup of, or
a conjugate of $G$, and thus must be strictly larger among the possibilities
provided by Guichard’s list. ∎

Theorem 1.1 is then proven by repeated applications of the bending
constructions as follows.

###### Proof of 1.1..

We let $\rho:\pi_{1}(S)\to\operatorname{SL}(n,\mathcal{O}_{K})$ be a
representation on the Hitchin component with Zariski closure
$G\neq\operatorname{SL}(n,\R)$. By the various methods in §4 we can construct
a matrix $A\in\operatorname{SL}(n,\mathcal{O}_{K})$ satisfying the following
properties:

  1. (a)

$A$ centralizes $\rho(\gamma)$ for some nonseparating simple closed curve
$\gamma$ (after possibly passing to a finite sheeted cover $\widetilde{S}\to
S$);

  2. (b)

$A\notin G$;

  3. (c)

$A=\exp(X)$ for some diagonalizable $X\in\mathfrak{sl}(n,\R)$.

From such a matrix, property (a) allows us to form the new bent representation
$\rho^{A}$. By property (b), along with Lemma 3.1, this also guarantees that
$\rho^{A}(\pi_{1}(\widetilde{S}))$ has a larger Zariski closure than $G$.
$\rho^{A}$ still remains on the Hitchin component of $\widetilde{S}$ by
property (c) as outlined at the beginning of this section.

We can then repeat this process (at most twice, when $n\neq 7$) until all of
the finitely many possible Zariski closures of Theorem 2.3 are ruled out
except for $\operatorname{SL}(n,\R)$ itself at which point, the final
representation must be Zariski dense, $K$-integral and on the Hitchin
component. ∎

##  4\. Constructing the bending matrices

To complete the proof of Theorem 1.1, it remains to be shown how one may
construct the matrices $A$ satisfying each of properties (a)–(c) used in the
proof. Different techniques for constructing such a matrix are needed
depending on the various possibilities of the initial Zariski closure provided
by Theorem 2.3 as well as the properties of the number field $K$. Before we
discuss the specific methods used to construct $A$ in each case, we record the
following pair of useful lemmas here which will be used throughout.

######  Lemma 4.1.

Given $M\in\operatorname{SL}(n,K)$ whose characteristic polynomial is in
$\mathcal{O}_{K}[t]$, then there is some $j>0$ so that
$M^{j}\in\operatorname{SL}(n,\mathcal{O}_{K})$.

###### Proof.

Using the rational canonical form of $M$, there is a
$P\in\operatorname{GL}(n,K)$ so that $M=PRP^{-1}$ where
$R\in\operatorname{SL}(n,\mathcal{O}_{K})$. Already here, we are implicitly
using the hypothesis that $K$ have class number one to guarantee that
$R\in\operatorname{SL}(n,\mathcal{O}_{K})$. Namely, if
$f\in\mathcal{O}_{K}[t]$ denotes the characteristic polynomial of $M$, then a
factorization of $f$ into irreducibles over $\mathcal{O}_{K}$ will be a
factorization into irreducibles over $K$ by Gauss’s lemma as $\mathcal{O}_{K}$
is a UFD. Thus the invariant factors of $f$ can be guaranteed to all be in
$\mathcal{O}_{K}[t]$ as well and so the rational canonical form of $f$ will
have entries in $\mathcal{O}_{K}$.

For any $N\in\mathcal{O}_{K}$, we consider the reduction map
$\operatorname{SL}(n,\mathcal{O}_{K})\to\operatorname{SL}(n,\mathcal{O}_{K}/(N))$.
$\operatorname{SL}(n,\mathcal{O}_{K}/(N))$ is finite so there is some $j$ so
that $R^{j}\equiv I\pmod{(N)}$. In other words, $R^{j}=I+NX$ for some
$X\in\operatorname{M}_{n}(\mathcal{O}_{K})$. In this case, we have that

$M^{j}=P(I+NX)P^{-1}=I+NPXP^{-1}.$

In the above expression, the only term depending on $N$ (other than $N$
itself) is $X$, which already has integral entries. Thus, if we pick
$N\in\mathcal{O}_{K}$ so that it clears all the denominators of all entries in
$P$ and $P^{-1}$, we get that $M^{j}\in\operatorname{SL}(n,\mathcal{O}_{K})$.
∎

######  Lemma 4.2.

Suppose $\rho(\gamma)\in\operatorname{SL}(n,\mathcal{O}_{K})$ has
characteristic polynomial $f(t)$ where $f\in\mathcal{O}_{K}[t]$ is irreducible
over $\mathcal{O}_{K}$. Then there is some
$A\in\operatorname{SL}(n,\mathcal{O}_{K})$ centralizing $\rho(\gamma)$, not
preserving the form $J$ and so that $A=\exp(X)$ for some diagonalizable
$X\in\mathfrak{sl}(n,\R)$.

###### Proof.

As $f$ is irreducible over the UFD $\mathcal{O}_{K}$, it is irreducible over
$K$. Let $\alpha\in\overline{K}$ denote some root of $f$ and consider the
extension $K(\alpha)/K$. Note that $[K(\alpha):K]=n$ as $f$ must be the
minimal polynomial of $\alpha$. All the roots of $f$ must be distinct and real
as $\rho$ is on the Hitchin component (2.1) and so $K(\alpha)$ has at least
$n$ real embedding as we may extend the (fixed) real embedding of $K$ to one
of $K(\alpha)$ by sending $\alpha$ to any of the other real roots of $f$. As a
consequence, the unit group $\mathcal{O}_{K(\alpha)}^{\times}$ has rank $\geq
n-1$.

By diagonalizing $\rho(\gamma)$ over $\R$ and using the fact that it has
distinct real eigenvalues, one sees that the centralizer of $\rho(\gamma)$ in
$\operatorname{SO}(J;\R)$ has rank $\frac{n}{2}$. In comparison,
$\mathcal{O}_{K(\alpha)}^{\times}$ has rank $\geq n-1$. Then $n-1>\frac{n}{2}$
as long as $n>2$.

Note then that there is a $K$-algebra isomorphism

$K(\alpha)\cong K[\rho(\gamma)]$

sending $\alpha\mapsto\rho(\gamma)$ because both of the above algebras are
isomorphic to $K[t]/(f)$ for the monic irreducible $f\in\mathcal{O}_{K}[t]$.
Consider now, the image of $\mathcal{O}_{K(\alpha)}^{\times}$ in
$K[\rho(\gamma)]$. By the above rank considerations, there is some infinite
order $u\in\mathcal{O}_{K(\alpha)}^{\times}$ whose image, $A^{\prime}\in
K[\rho(\gamma)]$, satisfies the property that no power of $A^{\prime}$
preserves the form $J$. Further assume that $u>0$ by replacing $u$ with $-u$
if necessary. $A^{\prime}$ is a polynomial in powers of $\rho(\gamma)$ and so
$A^{\prime}$ itself will still centralize $\rho(\gamma)$.

At the moment, $A^{\prime}$ still has some possibly undesirable properties,
but we will modify it to construct the matrix $A$. First, note $A^{\prime}$
has characteristic polynomial in $\mathcal{O}_{K}[t]$: its characteristic
polynomial factors as $\prod_{i=1}^{n}(t-u^{\sigma_{i}})$ where the
$\sigma_{i}:K(\alpha)\to\mathbf{C}$ are the $K$-embeddings of $K(\alpha)$. The
determinant of $A^{\prime}$ at the moment is
$v=\operatorname{Norm}_{K(\alpha)/K}(u)\in\mathcal{O}_{K}^{\times}$, but we
may pass to a higher power of $A^{\prime}$ and rescale by some power of
$v^{-1}$ so that $A^{\prime}$ has determinant $1$. This power of $A^{\prime}$
then has characteristic polynomial in $\mathcal{O}_{K}[t]$ and is in
$\operatorname{SL}(n,K)$, hence by Lemma 4.1, some further suitable power will
have entries in $\mathcal{O}_{K}$. Taking $A=(A^{\prime})^{j}$ to be such a
large enough power, we get $A\in\operatorname{SL}(n,\mathcal{O}_{K})$.

This matrix $A=(A^{\prime})^{j}\in\operatorname{SL}(n,\mathcal{O}_{K})$ still
centralizes $\rho(\gamma)$ and does not preserve the form $J$ as no power of
$A^{\prime}$ preserves $J$ by construction. Furthermore, such a $A$ will be
the exponential of some diagonalizable matrix as we arranged for $u>0$ and as
$A^{\prime}$ is a polynomial in powers of $\rho(\gamma)$, all of which are the
exponential of some diagonal matrix by Theorem 2.1. ∎

######  Remark.

In the odd-dimensional case, the characteristic polynomials of the
$\rho(\gamma)$ are never irreducible over $\mathcal{O}_{K}$ as they always
have an eigenvalue of $1$, due to being on the Hitchin component. Nonetheless,
the same conclusion of this lemma can easily be adopted to the odd-dimensional
case if we find $\rho(\gamma)$ with characteristic polynomial of the form
$(t-1)f(t)$ for some $f(t)\in\mathcal{O}_{K}[t]$ irreducible over
$\mathcal{O}_{K}$ since we may conjugate $\rho(\gamma)$ to be in block
diagonal form with an $(n-1)\times(n-1)$-block along the diagonal. In this
case, applying this result to this block gives the bending matrix $A$ in the
odd-dimensional case.

This lemma provides sufficient conditions for which the bending matrix $A$
used in the proof of Theorem 1.1 will exist. This implies that to construct a
bending matrix, it suffices to find a nonseparating simple closed curve
$\gamma\in\pi_{1}(S)$ whose characteristic polynomials are of the form $f(t)$
or $(t-1)f(t)$ for some irreducible $f(t)$ in $\mathcal{O}_{K}[t]$. When such
$\gamma$ cannot be found, we then use the reducibility of the characteristic
polynomials to explicitly construct bending matrices we may use as well. The
methods used in constructing these matrices vary depending on what the input
to the bend is, particularly, what the number field $K$ is, and what the
Zariski closure of the initial representation $\rho(\pi_{1}(S))$ is.

###  4.1. Bending when $K\neq\mathbf{Q}$

The simplest case of the bending procedure occurs when $K$ is a proper
extension of $\mathbf{Q}$ with class number one. In this case, we are able to
prove the following strengthening of Theorem 1.1.

######  Theorem 4.3.

If $K\neq\mathbf{Q}$ and
$\rho:\pi_{1}(S)\to\operatorname{SL}(n,\mathcal{O}_{K})$ is a representation
on the Hitchin component whose image is not Zariski dense, then for any
nonseparating simple closed curve $\gamma\in\pi_{1}(S)$, there is an
$A\in\operatorname{SL}(n,\mathcal{O}_{K})$ centralizing $\rho(\gamma)$ so that
the bent representation $\rho^{A}$ is on the Hitchin component and has Zariski
dense image.

In comparison to Theorem 1.1, when $K\neq\mathbf{Q}$, we are able to ensure
the representation is Zariski dense while circumventing the need to pass to
any finite sheeted covers. The class number one hypothesis enables various
integrality properties to be preserved, but the proof of this result is
ultimately enabled by the presence of infinite order units in
$\mathcal{O}_{K}$ allowing one to construct the bending matrix $A$ explicitly.

###### Proof of 4.3.

For any nonseparating simple closed $\gamma\in\pi_{1}(S)$, we consider the
characteristic polynomial $f$, of $\rho(\gamma)$. If $f$ is irreducible, then
we take the matrix $A$ to be the one guaranteed by Lemma 4.2. As $A$ doesn’t
preserve the (unique) form $J$ which $\rho(\pi_{1}(S))$ must preserve (as in
Lemma 3.1), then $\rho^{A}(\pi_{1}(S))$ must have Zariski closure larger than
$\operatorname{SO}(J;\R)$ or $\operatorname{Sp}(J;\R)$ and so it must be
Zariski dense by Theorem 2.3.

Thus we may assume that $f$ is reducible over $\mathcal{O}_{K}$. It suffices
to assume that $f=f_{1}f_{2}$ for $f_{i}\in\mathcal{O}_{K}[t]$ of degree
$n_{1},n_{2}\geq 1$ respectively. Replacing $f_{1}$ and $f_{2}$ with $uf_{1}$
and $u^{-1}f_{2}$ for some unit $u\in\mathcal{O}_{K}^{\times}$ if necessary,
we may also assume that $f_{1}$ and $f_{2}$ are monic and have constant terms
$(-1)^{n_{1}}$ and $(-1)^{n_{2}}$ respectively. From the factorization of the
characteristic polynomial, there is some matrix $P\in\operatorname{GL}(n,K)$
so that

$P\rho(\gamma)P^{-1}=\begin{pmatrix}C_{1}\\\ &C_{2}\end{pmatrix}$

where for $i=1,2$, $C_{i}\in\operatorname{SL}(n_{i},\mathcal{O}_{K})$ has
characteristic polynomial $f_{i}$. Here, we are using the fact that the
factorization of $f$ is over $\mathcal{O}_{K}$ and that $K$ has class number
one to guarantee that the entries of the $C_{i}$ are indeed in
$\mathcal{O}_{K}$.

Now, as $K\neq\mathbf{Q}$, $[K:\mathbf{Q}]>1$. Assuming $K\subset\R$ also
forces $K$ to have $\geq 1$ real embeddings, and together, this forces
$\mathcal{O}_{K}^{\times}$ to have rank $\geq 1$. We may thus fix an infinite
order unit $u\in\mathcal{O}_{K}^{\times}$ and, replacing $u$ with $-u$ if
necessary, we may also assume that $u>0$.

In this case, we take the matrix

$A^{\prime}:=P^{-1}\begin{pmatrix}u^{n_{2}}I_{n_{1}}\\\
&u^{-n_{1}}C_{2}\end{pmatrix}P.$

$A^{\prime}\in\operatorname{SL}(n,K)$ has characteristic polynomial still in
$\mathcal{O}_{K}[t]$ and clearly centralizes $\rho(\gamma)$ due to
$PA^{\prime}P^{-1}$ having the same block-diagonal structure as
$P\rho(\gamma)P^{-1}$. As $u>0$ and $C_{2}$ is diagonalizable,
$A^{\prime}=\exp(X)$ for some diagonalizable $X\in\sl(n,\R)$ as well. Note as
well that $A^{\prime}$ does not preserve the form $J$ as its eigenvalues are
$u^{n_{2}}$ (with multiplicity $n_{1}$) and $u^{-n_{1}}\lambda$ where
$\lambda$ is any of the (distinct, real) eigenvalues of $C_{2}$, and these
eigenvalues do not pair up as needed for any matrix preserving the form $J$.

Passing to a higher power of $A^{\prime}$ by Lemma 4.1,
$A=(A^{\prime})^{j}\in\operatorname{SL}(n,\mathcal{O}_{K})$ for sufficiently
large $j$ still centralizes $\rho(\gamma)$ and remains the exponential of some
diagonalizable matrix. As $u$ is an infinite order unit, $A$ will still fail
to preserve the form $J$ as $A^{\prime}$ did. Therefore, by Lemma 3.1, the
bent representation $\rho^{A}$ has Zariski dense image. ∎

By handling the $K\neq\mathbf{Q}$ case separately using the presence of
infinite order units in $\mathcal{O}_{K}$, we may then assume that
$K=\mathbf{Q}$ and $\mathcal{O}_{K}=\mathbf{Z}$ for the remainder of this
section.

###  4.2. Bending out of a principal $\operatorname{SL}(2,\R)$

When $K=\mathbf{Q}$, no infinite order units are present and so an alternate
method is necessary. The overall construction of the bending matrices here is
then a two-fold process: one to bend the representation out of a principal
$\operatorname{SL}(2,\R)$ and a second to bend the representation out of
either $\operatorname{Sp}(2k,\R)$ or $\operatorname{SO}(k+1,k)$ to finally get
a representation with Zariski dense image in $\operatorname{SL}(n,\R)$.
Different methods are used for each bend, the simpler one when the Zariski
closure of $\rho(\pi_{1}(S))$ is “as small as possible.”

By Theorem 2.3, the smallest possible Zariski closure that our representation
might have is that of a principal $\operatorname{SL}(2,\R)$ inside of
$\operatorname{SL}(n,\R)$. So suppose that
$\rho:\pi_{1}(S)\to\operatorname{SL}(n,\mathbf{Z})$ has Zariski closure
conjugate to $\tau_{n}(\operatorname{SL}(2,\R))$. In this case, we start by
fixing any nonseparating simple closed curve $\gamma\in\pi_{1}(S)$.

By Theorem 2.1, $\rho(\gamma)$ is diagonalizable with distinct real
eigenvalues. Moreover, as $\rho(\pi_{1}(S))$ is contained in some principal
$\operatorname{SL}(2,\R)$, basic properties about the irreducible
representations of $\operatorname{SL}(2,\R)$ tell us that the eigenvalues of
$\rho(\gamma)$ are of the form $\lambda^{n-2i-1}$ for $i=0,\ldots,n-1$. In
this case, the characteristic polynomial of $\rho(\gamma)$ is of the form
$(t-1)f(t)$ or $f(t)$ for some $f\in\mathbf{Z}[t]$ depending on the parity of
$n$. If $f(t)$ is irreducible in either of the above cases, we again apply
Lemma 4.2 to construct the matrix $A$ (in the $n$ odd case, we proceed as in
the remark after Lemma 4.2). Consequently, the matrix $A$ provided by this
theorem does not preserve the form this principal $\operatorname{SL}(2,\R)$
does, and so $\rho^{A}$ will then have Zariski dense image. So just as with
the $K\neq\mathbf{Q}$ construction, we are able to guarantee the
representation has Zariski dense image after a single bend, if we can find a
$\gamma\in\pi_{1}(S)$ with this property.

What remains in this case is when $f(t)$ is reducible over $\mathbf{Z}$. That
is, we may assume the characteristic polynomial of $\rho(\gamma)$ admits a
factorization of the form

$(t-1)f_{1}(t)f_{2}(t)\quad\textrm{or}\quad f_{1}(t)f_{2}(t),$

for polynomials $f_{1},f_{2}\in\mathbf{Z}[t]$ of degree $n_{1},n_{2}\geq 1$
respectively. From the factorization of the characteristic polynomial, it
follows then that there is some matrix $P\in\operatorname{GL}(n,\mathbf{Q})$
so that

$P\rho(\gamma)P^{-1}=\begin{cases}\displaystyle\begin{pmatrix}1\\\ &C_{1}\\\
&&C_{2}\end{pmatrix}&\textrm{if $n$ is odd}\\\
\displaystyle\begin{pmatrix}C_{1}\\\ &C_{2}\end{pmatrix}&\textrm{if $n$ is
even,}\end{cases}$

where for $i=1$ or $2$, $C_{i}\in\operatorname{SL}(n_{i},\mathbf{Z})$ has
characteristic polynomial $f_{i}$. We now take $A^{\prime}$ to be the matrix

$A^{\prime}=\begin{cases}\displaystyle P^{-1}\begin{pmatrix}1\\\ &I_{n_{1}}\\\
&&C_{2}\end{pmatrix}P&\textrm{if $n$ is odd}\\\ \displaystyle
P^{-1}\begin{pmatrix}I_{n_{1}}\\\ &C_{2}\end{pmatrix}P&\textrm{if $n$ is
even.}\end{cases}$

Note that in both cases, $\det(A^{\prime})=\det(C_{2})=1$ and $A^{\prime}$ is
in the $K$-centralizer of $\rho(\gamma)$.

By Lemma 4.1 there is some $j>0$ so that $(A^{\prime})^{j}$ has entries in
$\mathbf{Z}$. We take $A=(A^{\prime})^{j}$. Then by construction,
$A\in\operatorname{SL}(n,\mathbf{Z})$ centralizes $\rho(\gamma)$. Moreover,
$A$ is not contained in a principal $\operatorname{SL}(2,\R)$ because the
eigenvalues of $A$ are not of the form $\mu^{n-2i-1}$ for some $\mu\in\R$
since $A$ has an eigenvalue $1$ of multiplicity $>1$. Finally, we show that
$A$ is in the image of the exponential map. For this, it suffices to show
$A^{\prime}=\exp(X)$ for some $X\in\mathfrak{sl}(n,\R)$ since then,
$A=(A^{\prime})^{j}=\exp(jX)$. In this case, if we let
$\lambda_{1},\ldots,\lambda_{n_{2}}$ be the distinct positive real eigenvalues
of $C_{2}$, then $C_{2}$ is diagonalizable over $\R$ and we have that

$A^{\prime}=Q^{-1}\exp\begin{pmatrix}0\\\ &\ddots\\\ &&0\\\
&&&\log(\lambda_{1})\\\ &&&&\ddots\\\
&&&&&\log(\lambda_{n_{2}})\end{pmatrix}Q$

for some $Q\in\operatorname{GL}(n,\R)$. Moreover, $\det(C_{2})=1$ so that the
trace of the above matrix is $0$, hence $A^{\prime}=\exp(X)$ for some
diagonalizable $X\in\mathfrak{sl}(n,\R)$.

As a result, the matrix $A$ centralizing $\rho(\gamma)$ can be used to bend
the representation $\rho$ and produce a representation
$\rho^{A}:\pi_{1}(S)\to\operatorname{SL}(n,\mathbf{Z})$ on the Hitchin
component with Zariski closure strictly larger than a principal
$\operatorname{SL}(2,\R)$. At this point, the resulting bent representation
may still have Zariski closure conjugate to $\operatorname{SO}(k+1,k)$ or
$\operatorname{Sp}(2k,\R)$ when $n=2k+1$ or $n=2k$ respectively. Thus another
bend is still necessary to guarantee that one may produce a final
representation with Zariski dense image.

###  4.3. Bending out of $\operatorname{Sp}(2k,\R)$

We handle the even case first which follows the same general strategy as the
bend out of $\operatorname{SO}(k+1,k)$, but is practically easier, due to
facts about symplectic groups which simplify various steps. Assume that
$\rho:\pi_{1}(S)\to\operatorname{SL}(2k,\mathbf{Z})$ has Zariski closure
$\operatorname{Sp}(2k,\R)$. The starting point is the following Strong
Approximation result from the theory of algebraic groups.

######  Theorem 4.4 (Matthews et al., [MVW84]).

Let $G$ be a connected simply-connected absolutely almost simple algebraic
group defined over $\mathbf{Q}$ and $\Gamma\leqslant G(\mathbf{Q})$ a finitely
generated Zariski dense subgroup.

Then for all but finitely many primes $p$, the reduction map

$\pi_{p}:\Gamma\to G(\mathbf{F}_{p})$

is onto.

We will also need the following result of Borel’s.

######  Theorem 4.5 (Borel).

Let $p>4$ and let $R_{p}(2k)$ be the set of $2k\times 2k$ matrices in
$\operatorname{Sp}(2k,\mathbf{F}_{p})$ with reducible characteristic
polynomial. Then

$|R_{p}(2k)|\leq\left(1-\frac{1}{3k}\right)|\operatorname{Sp}(2k,\mathbf{F}_{p})|.$

By Theorem 4.4 for $G=\operatorname{Sp}(2k)$ and $\Gamma=\rho(\pi_{1}(S))$, we
obtain a surjection

$\pi_{p}\circ\rho:\pi_{1}(S)\twoheadrightarrow\operatorname{Sp}(2k,\mathbf{F}_{p}),$

for some prime $p$ which, when combined with Theorem 4.5, implies there exists
some loop $\eta\in\pi_{1}(S)$ so that the characteristic polynomial of
$\pi_{p}(\rho(\eta))$ is irreducible modulo $p$. Consequently, $\rho(\eta)$
has characteristic polynomial irreducible over $\mathbf{Z}$. Now, at the
moment, this loop may not be simple, but the following result allows us to
remedy this.

######  Theorem 4.6 (Scott, [Sco78]).

Every closed curve on $S$ lifts to a simple closed curve in some finite-
sheeted cover.

We now pass to a finite sheeted cover $\widetilde{S}\to S$ where $\eta$ lifts
simply and restrict our representation $\rho$, to the subgroup
$\pi_{1}(\widetilde{S})\leqslant\pi_{1}(S)$. By passing to a further finite
sheeted cover if needed, we may also assume that the loop
$\eta\in\pi_{1}(\widetilde{S})$ is nonseparating. The following lemma is
needed before then performing the bend out of $\operatorname{Sp}(2k,\R)$.

######  Lemma 4.7.

The subgroup $\rho(\pi_{1}(\widetilde{S}))\leqslant\rho(\pi_{1}(S))$ has
Zariski closure $\operatorname{Sp}(2k,\R)$.

###### Proof.

Let $H$ denote the Zariski closure of $\rho(\pi_{1}(\widetilde{S}))$. Clearly
$H\leqslant\operatorname{Sp}(2k,\R)$, but as $\widetilde{S}$ is still a closed
surface and $\rho|_{\pi_{1}(\widetilde{S})}$ lies on the Hitchin component of
$\widetilde{S}$, Guichard’s classification (2.3) implies that either
$H=\operatorname{Sp}(2k,\R)$ or $H$ is a principal $\operatorname{SL}(2,\R)$.

But $H$ is of finite index in $G$, so if $g_{1},\ldots,g_{k}$ are finitely
many coset representatives for $G/H$, then $g_{1}H\cup\ldots\cup g_{k}H$ is a
Zariski-closed subset containing $G$. If $H$ was a principal
$\operatorname{SL}(2,\R)$, this would imply that $\operatorname{Sp}(2k,\R)$ is
a union of finitely many principal $\operatorname{SL}(2,\R)$’s and simply for
dimension reasons this is even impossible. ∎

By this lemma, we have a representation
$\rho:\pi_{1}(\widetilde{S})\to\operatorname{SL}(n,\mathbf{Z})$ which is
Zariski dense in $\operatorname{Sp}(2k,\R)$ and on the Hitchin component of
$\widetilde{S}$. Moreover, the curve $\eta\in\pi_{1}(S)$ lifts simply in
$\widetilde{S}$ and the characteristic polynomial of $\eta$ is irreducible
over $\mathbf{Z}$. In light of Lemma 4.2, there is some bending matrix
$A\in\operatorname{SL}(n,\mathbf{Z})$ so that the bent representation
$\rho^{A}$ is Zariski dense in $\operatorname{SL}(n,\R)$.

###  4.4. Bending out of $\operatorname{SO}(k+1,k)$

In the odd $n$ case, the only possible representations remaining are the
$\mathbf{Q}$-integral ones with Zariski closure conjugate to
$\operatorname{SO}(k+1,k)$. Suppose that
$\rho:\pi_{1}(S)\to\operatorname{SL}(2k+1,\mathbf{Z})$ has Zariski closure
$\operatorname{SO}(J;\R)$ where $J$ is some symmetric bilinear form of
signature $(k+1,k)$. In this case, we prove the following fact.

######  Lemma 4.8.

There exists a $\eta\in\pi_{1}(S)$ so that $\rho(\eta)$ has characteristic
polynomial $(t-1)f(t)\in\mathbf{Z}[t]$ with $f(t)$ irreducible over
$\mathbf{Z}$.

The required loop is constructed in essentially the same manner as done in
§$3.2$ of [LT22], but we recap the construction, and note the minor
differences here. Again, the first step will be to look at the reduction map
$\operatorname{SO}(J,\mathbf{Z})\to\operatorname{SO}(J,\mathbf{F}_{p})$ for
some prime $p$. In this case, over odd-dimensional vector spaces over finite
fields, there is a unique orthogonal group up to isomorphism, dependent only
on the dimension and the prime $p$ (Theorem $5.8$ in [Suz82]). Thus we will
make the identification
$\operatorname{O}(J,\mathbf{F}_{p})=\operatorname{O}(2k+1,p)$. In this case,
we let $\operatorname{SO}(2k+1,p)$ denote the (unique up to isomorphism)
special orthogonal group over $\mathbf{F}_{p}$ and also set
$\Omega(2k+1,p)=[\operatorname{O}(2k+1,p),\operatorname{O}(2k+1,p)]\leqslant\operatorname{SO}(2k+1,p)$
to be the commutator subgroup. $\Omega(2k+1,p)$ is a simple subgroup of index
$4$ in $\operatorname{O}(2k+1,p)$ (cf. [Suz82], p. 383–384).

Just as in §4.3, the curve in Lemma 4.8 with the required characteristic
polynomial will be constructed by looking at the characteristic polynomials
modulo some prime $p$. However, we can’t simply apply Strong Approximation as
is stated in Theorem 4.4 simply due to the fact that the algebraic group
$\operatorname{SO}(J)$ is not simply connected. Instead, the following
corollary of Strong Approximation (cf. [Wei84]) still provides us with enough
in the image of the reduction map to construct the needed $\eta$.

######  Theorem 4.9.

For all but finitely many primes $p$, the image of the composition

$\pi_{p}\circ\rho:\pi_{1}(S)\to\operatorname{SO}(2k+1,p)$

contains the subgroup $\Omega(2k+1,p)$.

Lemma 4.8 then follows immediately from the previous theorem and the following
result whose proof is Theorem $3.8$ of [LT22].

######  Lemma 4.10.

For every prime $p$, there is a matrix in $\Omega(2k+1,p)$ with a
characteristic polynomial of the form $(t-1)\overline{f}(t)$ where
$\overline{f}$ is irreducible modulo $p$.

By these last two results, we may find some $\eta\in\pi_{1}(S)$ so that
$\pi_{p}(\rho(\eta))$ has characteristic polynomial of the form
$(t-1)\overline{f}(t)$ for some $\overline{f}$ irreducible modulo $p$. It
follows that $\rho(\eta)$ has characteristic polynomial of the form
$(t-1)f(t)$ for some $f$ irreducible over $\mathbf{Z}$. As before, $\eta$ may
not be simple, but Theorem 4.6 again lets us pass to a finite sheeted cover
$\widetilde{S}\to S$ where $\eta$ lifts simply to a nonseparating curve.
Again, as in Lemma 4.7,
$\rho(\pi_{1}(\widetilde{S}))\leqslant\rho(\pi_{1}(S))$ also has Zariski
closure $\operatorname{SO}(J;\R)$ and so we may perform the bend on the
restriction $\rho:\pi_{1}(\widetilde{S})\to\operatorname{SL}(n,\mathbf{Z})$,
along the simple closed curve $\eta$, to get a representation
$\rho^{A}:\pi_{1}(\widetilde{S})\to\operatorname{SL}(n,\mathbf{Z})$ with
Zariski dense image in $\operatorname{SL}(n,\R)$ via Lemma 4.2. This last
remaining case then completes the proof of Theorem 1.1.

##  5\. Applications

The methods of this paper also have a number of useful consequences and
applications. We record some of these here.

######  Corollary 5.0.1.

If $\operatorname{Hit}_{n}(S)$ contains an $\mathcal{O}_{K}$-point, then there
are infinitely many $\mathcal{O}_{K}$-points in $\operatorname{Hit}_{n}(S)$
which are Zariski-dense.

###### Proof.

This mainly is a consequence of the fact that the construction of the bending
matrices $A$ rely on a number of arbitrary choices made, and any difference in
these choices lead to different Zariski dense representations of $\pi_{1}(S)$.
For instance, if $A$ is a bending matrix satisfying properties (a)–(c) used in
the proof of Theorem 1.1, then so is $A^{m}$ for any integer $m\geq 1$, and
hence $\rho^{A^{m}}$ are all alternative bends of $\rho$ with Zariski dense
image. Pairwise non-conjugacy of this family of Zariski dense representations
also follows by an argument similar to the one given in Lemma 3.1 using
Schur’s lemma, leveraging the fact that the family of bends
$\\{\rho^{A^{m}}\\}_{m=1}^{\infty}$ all agree on the subgroup of the cut
subsurface, $\pi_{1}(S^{\prime})$, whose Zariski closure contains a principal
$\operatorname{SL}(2,\R)$.

Additionally, in the setting of §4.1, the construction works for any
nonseparating simple closed curve on $S$, and so for each curve, different
Zariski dense surface groups may be produced. Notably, all these
representations also lie in different mapping class group orbits in
$\operatorname{Hit}_{n}(S)$ as they all have distinct images. ∎

######  Corollary 5.0.2.

The Zariski dense representations in $\operatorname{Hit}_{n}(S)$ are
(classically) dense in the Hitchin component.

###### Proof.

This rests on the fact that the same techniques used in §4.1 may be used to
perform a bend. However, in this case, when we no longer require anything
regarding integrality of the representation to be preserved, we may take
bending matrices of the form $\exp(tX)$ for as small of a $t>0$ as we desire.
Furthermore, these bends can be performed about any simple closed curve on
$S$.

Thus for any $\rho\in\operatorname{Hit}_{n}(S)$ and any open neighborhood of
$\rho$, we may bend about any simple closed curve using a matrix $\exp(tX)$
for as small $t>0$ as necessary to produce Zariski dense representations near
$\rho$. ∎

Other interesting questions in these directions still remain. For instance,
whether or not the representations produced by this construction are non-
commensurable (up to conjugacy) seems to be unknown. Part of this question is
answered in [LR16] for representations into cocompact lattices of
$\operatorname{SL}(3,\R)$. There, the authors deduce pairwise non-
commensurability up to conjugacy of an infinite family of thin surface
subgroups by proving that the surface subgroups in this family had
projectively distinct limit sets.

In this context, the associated limit set may be studied in terms of the
hyperconvex Frenet curve
$\xi_{\rho}:\partial_{\infty}\pi_{1}(S)\to\operatorname{Flag}(\R^{n})$
associated to any Hitchin representation $\rho$, first introduced in [Lab06].
At the moment, we can show that the family of bends
$\\{\rho^{A^{m}}\\}_{m=1}^{\infty}$ produces representations whose flag curves
are distinct, i.e. the functions $\xi_{\rho^{A^{j}}}$ and $\xi_{\rho^{A^{k}}}$
are distinct whenever $j\neq k$. Full non-commensurability up to conjugacy
would require showing that these flag curves have distinct images, up to the
action of $\operatorname{SL}(n,\R)$, which is more subtle and seems to be
currently unknown.

We close with some known examples of representations this bending construction
can be applied to.

  1. (1)

The $(3,4,4)$-triangle group, $\Delta(3,4,4)$, is the fundamental group of the
orbifold with signature $S^{2}(3,4,4)$. Using the presentation
$\Delta(3,4,4)=\langle\alpha,\beta,\gamma\,|\,\alpha^{3}=\beta^{4}=\gamma^{4}=\alpha\beta\gamma=1\rangle$,
the representation realizing the (unique up to conjugacy) hyperbolic structure
on this orbifold is given by:

$\displaystyle\rho(\alpha)$ $\displaystyle=\begin{pmatrix}0&-1\\\
1&\phantom{-}1\end{pmatrix},$ $\displaystyle\rho(\beta)$
$\displaystyle=\begin{pmatrix}0&-1-\sqrt{2}\\\
-1+\sqrt{2}&\sqrt{2}\end{pmatrix},$ $\displaystyle\rho(\gamma)$
$\displaystyle=\begin{pmatrix}1-\sqrt{2}&-\sqrt{2}\\\
-1+\sqrt{2}&-1\end{pmatrix}.$

As this comes from a hyperbolic structure on $\Delta(3,4,4)$, $\rho$
represents a point in $\mathcal{T}(S^{2}(3,4,4))$. In [LT22] (Theorem $2.1$),
the authors show that, when composing with the irreducible representation
$\tau_{n}:\operatorname{SL}(2,\R)\to\operatorname{SL}(n,\R)$, $\tau_{n}\rho$
can be conjugated to have image inside $\operatorname{SL}(n,\mathbf{Z})$ when
$n$ is odd, or $\operatorname{SL}(n,\mathbf{Z}[\sqrt{2}])$ when $n$ is even
(note: $\mathbf{Z}[\sqrt{2}]=\mathcal{O}_{\mathbf{Q}(\sqrt{2})}$ is a PID).

$\Delta(3,4,4)$ contains a torsion-free subgroup of finite-index which is the
fundamental group of some closed surface: $\pi_{1}(S)\leqslant\Delta(3,4,4)$.
Restricting this representation of $\Delta(3,4,4)$ to one of $\pi_{1}(S)$, we
see that Hitchin component of $S$ therefore contains a $\mathbf{Z}$-point or a
$\mathbf{Z}[\sqrt{2}]$-point depending on the parity of $n$, and so can be
bent to being Zariski dense.

  2. (2)

Examples of $\mathbf{Q}$-integral even-dimensional Hitchin representations are
largely unknown. At the moment of writing, such examples are only known when
$n=4$. Long and Thistlethwaite provided such examples of Hitchin
representations into $\operatorname{SL}(4,\mathbf{Z})$ in [LT18]: one coming
from an infinite family of representations of the triangle group
$\Delta(3,3,4)$, along with one of the triangle group $\Delta(2,4,5)$. These
methods then provide an alternate proof of the results of the aforementioned
paper, giving Zariski dense surface subgroups in
$\operatorname{SL}(4,\mathbf{Z})$, but additional examples for
$\operatorname{SL}(2k,\mathbf{Z})$ when $k>2$ are still unknown.

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