# The Lie bracket of undirected closed curves on a surface

Moira Chas  Stony Brook Mathematics Department and Institute for Mathematical
Sciences<EMAIL_ADDRESS>and  Arpan Kabiraj  Department of
Mathematics, Indian Institute of Technology Palakkad<EMAIL_ADDRESS>
###### Abstract.

A Lie bracket defined on the linear span of the free homotopy classes of
undirected closed curves was discovered in stages passing through Thurston’s
earthquake deformations, Wolpert’s corresponding calculations with Hamiltonian
vector fields and Goldman’s algebraic treatment of the latter leading to a Lie
bracket on the span of directed closed curves. The purpose of this work is to
deepen the understanding of the former Lie bracket which will be referred to
as the Thurston-Wolpert-Goldman Lie bracket or, briefly, the TWG bracket.

We give a local direct geometric definition of the TWG bracket and use this
geometric point of view to prove three results: firstly, the center of the
TWG-bracket is the Lie sub algebra generated by the class of the trivial loop
and the classes of loops parallel to boundary components or punctures;
secondly the analogous result hold for the centers of the universal enveloping
algebra and of the symmetric algebra determined by the TWG Lie algebra; and
thirdly, in terms of the natural basis, the TWG bracket of two non-central
curves is always a linear combination of non-central curves. We also give a
brief and more illuminating proof of a known result, namely, the TWG bracket
counts intersection.

We conclude by discussing substantial computer evidence suggesting an
unexpected and strong conjectural statement relating the intersection
structure of curves and the TWG bracket, namely, if the TWG bracket of two
distinct undirected curves is zero then these curve classes have disjoint
representatives.

The main tools are basic hyperbolic geometry and Thurston’s earthquake theory.

##  1\. Introduction

###  1.1.  Geometric definition of the Thurston-Wolpert-Goldman Lie algebra

Let $\Sigma$ be an oriented Riemann surface, not necessarily of finite type,
which carries a complete metric of constant curvature $-1$.

Given two undirected curves $x$ and $y$ on $\Sigma$ and a transversal
intersection point $P$ of $x$ and $y$, one can define a new curve
${(x*_{P}y)}_{0}$ as follows: orient $x$ and $y$ in such a way that the
orientation of the surface agrees with the orientation determined by the two
ordered oriented branches of $x$ and $y$ emanating from $P$, then cut the two
branches at $P$ and reconnect the curves following the new orientations (that
is, perform the usual loop product at $P$), and then discard the orientation
of the composed curve.

A second new curve ${(x*_{P}y)}_{\infty}$ is defined similarly as
${(x*_{P}y)}_{0}$ but orientating $x$ and $y$ so that the orientation
determined by the two oriented branches emanating from $P$ disagree with the
orientation of the surface. Figure 1 shows the two possible re-connections
around $P$.

Figure 1. Local picture of the two smoothings associated with an intersection
point $P$ between two undirected curves $x$ and $y$.

The free homotopy class of a curve $x$ on $\Sigma$ is denoted by
$\widetilde{x}$. (Unless explicitly noted, all curves considered are
undirected). Let $\mathbb{K}$ be a commutative ring containing the integers.
The set of free homotopy classes of undirected closed curves is denoted by
$\widetilde{\pi}$ and the free module generated by $\widetilde{\pi}$ over a
ring $\mathbb{K}$ is denoted by $\mathbb{K}\widetilde{\pi}$. We now give the
definition of a Lie algebra on $\mathbb{K}\widetilde{\pi}$, by giving the
bracket of a pair of elements of the basis $\widetilde{\pi}$, and then,
extending bilinearly.

Consider two curves $x$ and $y$ intersecting only in transversal double
points. We define geometrically $\left[\widetilde{x},\widetilde{y}\right]$ as
the sum over all the intersection points of $x$ and $y$ of the free homotopy
classes of two signed smoothings associated with each intersection point of
$x$ and $y$. In symbols,

$\left[\widetilde{x},\widetilde{y}\right]=\sum_{P\in x\cap
y}\widetilde{(x*_{P}y)}_{0}-\widetilde{(x*_{P}{y})}_{\infty}.$

Clearly, if $\widetilde{x}$ and $\widetilde{y}$ have disjoint representatives,
$[\widetilde{x},\widetilde{y}]=0$. We extend the bracket linearly to
$\mathbb{K}\widetilde{\pi}$ and call this structure, the _Thurston-Wolpert-
Goldman Lie algebra of undirected curves, or briefly, the TWG Lie algebra_.

######  Remark 1.1.

Note that the smoothings at an intersection point can be defined for each pair
of branches if the intersection is transversal, but the point could have
multiplicity larger than two. More about this in Subsection 2.1.

######  Remark 1.2.

The definition of the Goldman Lie algebra of directed curves in [11] was
geometric. However the TWG Lie algebra of undirected curves, also defined in
[11] is defined, using an algebraic argument, as a sub Lie algebra of the
Goldman Lie algebra of directed curves. See Section 7 for details.

Wolpert [25, Theorem 4.8], discovered a sub-Lie algebra of vector fields, the
twist lattice, on the linear span of the Fenchel-Nielsen vector fields
associated to curves (which are the infinitesimal generators of the
earthquakes along these curves) and gave a topological description of this Lie
algebra. Goldman [11] showed that the twist lattice Lie algebra is the
homomorphic image of a more basic Lie algebra, the TWG-Lie algebra. Namely, he
proved [11, Theorem 5.2 and §5.12] that the TWG bracket (defined above) is
skew-symmetric and satisfies the Jacobi identity. According to Goldman, the
embedding of the TWG Lie algebra in the Goldman Lie algebra -which he used to
define the TWG algebra - was first observed by Dennis Johnson.

The TWG Lie algebra, and its “cousin”, the Goldman Lie algebra (see Section 7
for a definition) are infinite-dimensional and still have many mathematical
“secrets” to reveal. In this work, we make extensive use of hyperbolic
geometry to make these Lie algebras “talk” about their secrets.

###  1.2. Main results, road maps for the proofs and an unexpected conjecture

####  1.2.1. The Center of the TWG Lie algebra

Recall that the center of a Lie algebra on $V$ is the set of elements $v\in V$
such that $[v,w]=0$ for all $w\in V$. Etingof in [10] proved using
representation theory that the center of the Goldman Lie algebra of directed
curves on a closed surface and coefficients in $\mathbb{C}$ is generated by
the (class of the) trivial loop. It is not hard to extend Etingof’s result to
other rings containing $\mathbb{Z}$. The second author in his Ph.D. thesis,
determined that the center of the Goldman Lie algebra of curves on surfaces
with boundary is generated by the trivial loop together with all curves
parallel to the boundary components [14]. His proof treats all cases (closed
surfaces or with boundary) geometrically.

In Section 3, using geometric methods, we study the center of the TWG Lie
algebra of undirected curves:

######  Theorem (Center).

The center of the TWG Lie algebra of curves is linearly generated by the class
of curves homotopic to a point, and the classes of curves winding multiple
times around a single puncture or boundary component.

A difficulty that arises in the study of the center using topological or
geometric tools is that formal linear combinations of classes of curves, (as
opposed to single classes of curves) have to be considered. Thus, the
characterization of the center requires more argument than that of the
Counting Intersection Theorem below.

Here is the idea of the proof of the Center Theorem: The intersection points
of a union of geodesics $y_{1},y_{2},...,y_{k}$ and different powers of a
simple geodesic $x$ are “kind of ” the same when the powers vary -they are the
same “physical points” and hence the angles are the same. If the bracket of
$x^{m}$ with a linear combination of $y_{1},y_{2},...,y_{k}$ is zero for
enough values of $m$, then there are pairs of intersection points of $x$ with
the union of $y_{1},y_{2},...,y_{k}$ that yield terms of the corresponding
brackets that cancel for different values of $m$. This implies pairs of angles
at intersection points are supplementary or congruent for all metrics. By
twisting along the simple curve $x$, and some hyperbolic geometry we show that
both possibilities lead to a contradiction.

####  1.2.2. The Centers of the Universal Enveloping algebra and of the
Symmetric Algebra of the TWG Lie Algebra

In Section 4, we extend the method described in the above paragraph further to
study the universal enveloping algebra
$\mathcal{U}(\mathbb{K}\widetilde{\pi})$ and the symmetric algebra
$\mathcal{S}(\mathbb{K}\widetilde{\pi})$ of the TWG Lie algebra. The universal
enveloping algebra $\mathcal{U}(\mathbb{K}\widetilde{\pi})$ admits a natural
Poisson algebra structure where the Poisson bracket is the commutator. The
symmetric algebra $\mathcal{S}(\mathbb{K}\widetilde{\pi})$ admits natural
Poisson algebra structure induced from the TWG-Lie bracket using Leibniz rule,
(see Section 4 for definitions and references.) Using our geometric
computations together with Poincaré-Birkhoff-Witt theorem, we compute the
Poisson center of the each of the Poisson algebras
$\mathcal{U}(\mathbb{K}\widetilde{\pi})$ and
$\mathcal{S}(\mathbb{K}\widetilde{\pi})$.

######  Theorem (Poisson Center).

The Poisson centers of the Poisson algebras
$\mathcal{U}(\mathbb{K}\widetilde{\pi})$ and
$\mathcal{S}(\mathbb{K}\widetilde{\pi})$ are generated by scalars
$\mathbb{K}$, the free homotopy class of constant curve and the curves
homotopic to boundary and punctures.

Let $\Sigma$ be a closed surface. Consider $\mathcal{M}_{\mathbb{C}}^{n}$, the
algebraic variety associated to the moduli space of representations of
$\pi_{1}(\Sigma)$ into $GL(n,\mathbb{C})$ up to conjugation, which on its
smooth part admits the Goldman symplectic structure (see [12], [2]).

In [11], Goldman defined the homomorphism of Poisson algebras
$\mathcal{S}(\mathbb{C}{\pi})\rightarrow\mathbb{C}[\mathcal{M}_{\mathbb{C}}^{n}]$
defined by $\Phi_{n}({x})=tr_{x}$ (here, $\mathbb{C}{\pi}$ denotes the module
spanned by the set $\pi$ of conjugacy classes of fundamental group of surface
- that is, the set of free homotopy classes of directed curves- and
$\mathcal{S}(\mathbb{C}{\pi})$ denotes the symmetric algebra of
$\mathbb{C}{\pi}$). The map $\Phi_{n}$ is surjective [10] but not injective in
general. However Etingof [10, Proposition 2.2] proved that given any finite
dimensional subspace $V$ of $\mathbb{C}\pi$, there exists $N$ such that
$\Phi_{n}|_{V}$ is injective for all $n\geq N$. Using this result together
with the fact that Poisson center of
$\mathbb{C}[\mathcal{M}_{\mathbb{C}}^{n}]$ being $\mathbb{C}$, Etingof
computed the center of $\mathbb{C}\pi$. In principle Etingof’s method could be
used to compute the center of $\mathbb{C}\widetilde{\pi}$ by replacing
$GL_{n}(\mathbb{C})$ in the definition of $\mathcal{M}_{\mathbb{C}}^{n}$ by
one of the following groups: $O_{n}(\mathbb{C}),Sp_{n}(\mathbb{C})$ or
$U_{n}(\mathbb{C})$ [11, Theorem 3.14 and Theorem 5.13]. However, as with the
Etingof’s proof, this possible method will work only for closed surfaces. Our
proof, on the other hand, works for any complete hyperbolic surface.

Chas and Gadgil [7] used geometric group theory to study the quasi-geodesic
nature of the lifts of the terms of the Goldman bracket. Using this result and
the facts that lifts of a simple closed curve are disjoint, in [14], the
second author computed the center of $\mathbb{K}\pi$. Although, this method
could be used to compute the center of $\mathbb{K}\widetilde{\pi}$, there are
two drawbacks. Firstly because the proof is based on case by case
considerations, it is long, technical and geometrically less transparent than
our proof here. Secondly because of the geometric group theory techniques, the
various bounds obtained would be qualitative and not quantitative.

####  1.2.3. Canonical decomposition of the Goldman Lie algebra and the TWG
Lie algebra

Denote by C (resp. $\widetilde{\text{C}}$) the free homotopy classes of the
directed (resp. undirected) trivial curve, and the curves or powers of curves
parallel to boundary components. (Note that this C and. $\widetilde{\text{C}}$
are a basis of the center of the Goldman and TWG Lie algebras respectively).
Denote by $\pi_{0}$ (resp. $\widetilde{\pi}_{0}$) the set of free homotopy
classes of directed (resp. undirected) closed curves minus C (resp.
$\widetilde{\text{C}}$).

Goldman [11] stated that for closed surfaces, the Goldman Lie algebra admits a
canonical decomposition $\mathbb{Z}\text{O}\oplus\mathbb{Z}{\pi}_{0}$, where O
represents the class of constant curve. The result is valid, but the proof has
a gap, namely, it is not true that if $\alpha$ is the class of a directed
curve, and $\bar{\alpha}$ is the class of $\alpha$ with opposite direction,
then the Goldman bracket of $\alpha$ and $\bar{\alpha}$,
$[\alpha,\bar{\alpha}]$ is zero. For instance, if $\alpha$ is a figure eight
curve in the pair of pants (that goes around two boundary components) then
$[\alpha,\bar{\alpha}]$ has two terms that do not cancel. On the contrary, the
first author has conjectured in [5] that for all directed curves $\beta$ the
number of terms of $[\beta,\bar{\beta}]$ (counted with multiplicity) is twice
the self-intersection number of $\beta$. This conjecture is a “pre-theorem” in
the sense that it is strongly supported by computer evidence.

Our techniques yield a proof of Goldman’s direct sum statement about the Lie
algebra of directed curves, and also, the analogous result for the TWG
bracket.

######  Theorem (Canonical Decompositions).

The Goldman bracket of two non-central directed curve classes, expressed in
the natural basis, does not contain a central element as a term. The analogous
result holds for the TWG bracket.
(1) The Goldman Lie algebra admits a Lie algebra decomposition
$\mathbb{K}\mathrm{C}\oplus\mathbb{K}\pi_{0}$.
(2) The TWG Lie algebra also admits a Lie algebra decomposition
$\mathbb{K}\widetilde{\mathrm{C}}\oplus\mathbb{K}\widetilde{\pi}_{0}$.

####  1.2.4. A new proof of the Counting Intersection Theorem

Given two free homotopy classes of closed curves $\widetilde{x}$ and
$\widetilde{y}$, the _geometric intersection number of $\widetilde{x}$ and
$\widetilde{y}$_, denoted by $i(\widetilde{x},\widetilde{y})$, is defined to
be the smallest number of crossings of a pair of representatives
$\widetilde{x}$ and $\widetilde{y}$, that intersect in transversal double
points. Goldman [11, Theorem 5.7] proved that if the TWG bracket of two
classes of undirected curves is zero, and one of them has a simple
representative, then the classes contain disjoint representatives. He also
prove the analogous result for directed curves. Chas [6] using combinatorial
group theory generalized this result by proving that the Goldman and the TWG
bracket of two curves, one of them simple, counts intersection number. In both
cases, the main tool was the use of free products with amalgamation and HNN
structures on the fundamental group of the surface, determined by simple
curves. In the case of the TWG- bracket, Chas’s proof was complicated
requiring several cases and combinatorial technical lemmas. Our techniques
allow us to give a new geometric proof that the TWG bracket counts
intersections

######  Theorem (Counting Intersection).

If $x$ and $y$ are closed curves and $x$ is simple then the number of terms
(counted with multiplicity) of $[\widetilde{x},\widetilde{y}]$ is twice the
intersection number, $2i(\widetilde{x},\widetilde{y})$.

One noteworthy point about our geometric definition of the TWG Lie bracket
here and the resulting angle technique is a new proof in two straightforward
statements proving no cancellation of terms in the TWG bracket: The first of
these statements is that cancellation means that a certain pair of angles are
supplementary; the second, is that earthquaking along the simple curve changes
both angles strictly monotonically (both decreasing or both increasing). This
is a contradiction, proving the result.

####  1.2.5. A computational result and conjectural characterization of
disjointness of closed curves via the TWG bracket

The Counting Intersection Theorem is only valid if of the curves is simple,
see 5.1. However, computer experiments suggest that the TWG bracket actually
detects disjointness in all cases. Thus we conjecture:

######  Conjecture.

If the TWG Lie bracket of two distinct classes of undirected curves is zero,
then the classes have disjoint representatives.

The above statement was verified computationally for as many classes of curves
available computers can handle (see Section 5 for precise statements). This
conjecture is unexpected, given that the equivalent statement does not hold
for the Goldman Lie bracket on directed curves. In other words, there are
examples of pairs of classes of directed curves with Goldman bracket zero and
without disjoint representatives, see for instance Example 5.2.

There are also examples that show the TWG Lie bracket of two non-simple curves
can have cancellation, see Example 5.1. Thus, the hypothesis of one of the
curves being simple in the Counting Intersection Theorem cannot be dropped in
general.

Bracket |  $[x,y]=0$, $x$ simple |  $x$ simple then $[x,y]$ |  $[x,y]=0$ implies |  $[x,y]$ counts
---|---|---|---|---
| implies disjointness | counts intersection | disjointness | intersection
TWG | Yes | Yes | Yes, if conjecture holds | No
Goldman | Yes | Yes | No | No

####  1.2.6. Direct proof of the Jacobi identity for the TWG bracket

In Section 7 we prove that our definition and that of Goldman [12] coincide.
For completeness, in Section 6, we prove that the TWG bracket satisfies the
Jacobi identity.

###  1.3. Applications

In [23], Turaev introduced various skein modules associated to the set of
isotopy classes of links in the three manifold $\Sigma\times I$ for the
quantization of $\mathcal{U}(\mathbb{K}\widetilde{\pi})$ and
$\mathcal{S}(\mathbb{K}\widetilde{\pi})$. In [13], Hoste and Przytycki
independently gave another quantization of the same in terms of _homotopy
skein modules_. Our results about the center of
$\mathcal{U}(\mathbb{K}\widetilde{\pi})$ and
$\mathcal{S}(\mathbb{K}\widetilde{\pi})$ can be used to compute the center of
these skein modules. See [16].

###  1.4. Other results relating the intersection structure of curves with the
Goldman and TWG bracket

In [23], Turaev discovered cobracket operations which gives Goldman algebra
and TWG Lie algebra the structure of a Lie bialgebra. Trying to understand the
relation of the Goldman-Turaev Lie bialgebra with intersection and self-
intersection of curves on surfaces Chas and Sullivan discovered String
Topology [9], a structure that generalizes the Goldman Lie algebra and the
Turaev Lie coalgebra to arbitrary orientable manifolds of all dimensions.

Chas and Krongold [8] proved that, on a surface with boundary, a non-power
directed curve $x$ is simple if and only if the Goldman Lie bracket of $x$
with $x^{m}$ is non-zero, for any $m\geq 3$. Moreover, for any directed curve
$x$, the number of terms, counted with multiplicity, of the Goldman Lie
bracket of $x$ with $x^{m}$ is $2m$-times the geometric self-intersection
number of $x$. (Computer evidence suggests these statements are also true when
$m=2$. We are working on a proof of this result.)

Chas and Gadgil [7] proved that if $x$ and $y$ are non-power directed curves,
then for $p$ and $q$ large enough, the Goldman Lie bracket $[{x^{p}},{y^{q}}]$
counts the geometric intersection number of the classes of $x$ and $y$.

Cahn and Tchernov [4] determined that the Andersen-Mattes-Reshetikhin Poisson
bracket (a generalization of the Goldman Lie bracket) counts intersections of
two classes of curves, when the classes are distinct.

Kawazumi and Kuno [17] proved that the center of the Goldman Lie algebra on
directed curves on a surface with infinite genus and one boundary component is
spanned by constant loop and powers of loops parallel to the boundary
component.

Recently in [21], Turaev gave a new description of Goldman Lie algebra using
_star-fillings_ (also see [22]).

In both the Turaev cobracket [23] and the Chas-Sullivan [9] efforts the
motivation to understand embedded curves was related to the Jaco-Stallings
[20] sixties equivalence between a statement about embedded curves and the
Poincaré conjecture. Since this statement about embedded curves a five hundred
pages treatrise by Morgan and Tian [19] of Perelman’s work it is even more
compelling to give a geometric proof (hopefully, in substantially less than
five hundred pages) in the language of curves and surfaces. The papers
mentioned show this is a vast subject.

###  1.5. Organization of the paper

This work is organized as follows: In Section 2 we recall some results of
hyperbolic geometry and give the description of a lift of the terms of TWG Lie
bracket to the upper half plane ${\mathbb{H}}$. We also prove the Counting
intersection Theorem in Section 2. In Section 3 we prove the Center Theorem.
In Section 4 we compute the Poisson center of the universal enveloping algebra
and symmetric algebra of TWG Lie algebra. In Section 5 we give explicit
examples of computations of Goldman Lie bracket an TWG Lie bracket of curves
in some low complexity hyperbolic surfaces and state a conjecture about TWG
Lie bracket. In Section 7, we prove that our definition coincides with that of
Goldman [11]. In particular, this implies that the TWG algebra is well defined
on homotopy classes of undirected curves and it is indeed a Lie algebra. For a
proof of Jacobi’s identity for TWG bracket see Section 6. In Section 8 we
prove the Canonical Decomposition Theorem.

## Acknowledgements

This work benefited from communications with Vladimir Turaev, Scott Wolpert
and William Goldman. The first author was partially supported by the NSF. The
second author was supported by the DST, India: INSPIRE Faculty fellowship.

##  2\. Intersection points, angles and earthquakes. Proof of the Counting
Intersection Theorem

Denote by $\mathcal{T}$ the Teichmüller space associated with the surface
$\Sigma$. A closed curve on $\Sigma$ is an $X$-geodesic if it is a geodesic
for the metric $X\in\mathcal{T}$.

###  2.1. Intersection points and metrics

In order to be able to follow intersection points of two curves through
homotopies of these curves, we need to refine the definition of intersection
point: If $x$ and $y$ are two closed curves intersecting transversally, an
$(x,y)$-intersection point is a point $P$ on the intersection of $x$ and $y$,
together with a choice of a pair of small arcs, one of $x$ and the other of
$y$, intersecting only at $P$.

######  Remark 2.1.

For any two curves $x$, $y$ (possibly with intersection points of multiplicity
larger than two) we have

$i(\widetilde{x},\widetilde{y})=\mathrm{min}\\#\\{(x,y)\mbox{-intersection
points}:x\in\widetilde{x},y\in\widetilde{y}\mbox{, $x$ and $y$ intersect
transversally}\\}.$

Here the number of intersection points counted with multiplicity, namely $k$
lines intersecting transversally at a point counts as “$k$ choose two”.

By Thurston’s Earthquake Theorem (see the appendix of [18] for a proof), there
is a unique earthquake path between any pair $X,X^{\prime}$ of elements of
$\mathcal{T}$. The next lemma can be proved by following the
$(x,y)\mbox{-intersection points}$ of two $X$-geodesics $x$ and $y$ along this
unique geodesic path.

######  Lemma 2.2.

Let $X,X^{\prime}\in\mathcal{T}$ and let $x$ and $y$ be two $X$-geodesics. If
$x^{\prime}$, $y^{\prime}$ are two $X^{\prime}$-geodesics such that both pairs
$x,x^{\prime}$ and $y,y^{\prime}$ are homotopic then there is a canonical
bijection between the $(x,y)\mbox{-intersection points}$ and the
$(x^{\prime},y^{\prime})\mbox{-intersection points}$.

######  Remark 2.3.

Lemma 2.2 can be also proved using that $\mathcal{T}$ is simply connected,
instead of Thurston’s Earthquake Theorem.

###  2.2. Angles, metrics and earthquakes

Fix a metric $X\in\mathcal{T}$ and $P$, an $(x,y)$-intersection point of two
$X$-geodesics $x$ and $y$. For each metric $Y\in\mathcal{T}$, the _$Y$ \-
angle of $x$ and $y$ at $P$_, denoted by $\phi_{P}(Y)$, is defined as the
angle at the intersection point corresponding to $P$ by Lemma 2.2, of the two
$Y$-geodesics homotopic to $x$ and $y$, measured from the geodesic homotopic
to $y$ to the geodesic homotopic to $x$, following the orientation of the
surface (see Figure 2). Clearly, $\phi_{P}(X)$ is the angle at $P$, from $y$
to $x$ following the orientation of the surface. Observe that
$\phi_{P}(Y)\in(0,\pi)$. (The angle $\phi_{P}$ is defined from $y$ to $x$ as
is done in [24]).

Figure 2. The angle $\phi_{P}$

Following [18], for each simple $X$-geodesic $x$ and each real number $t$,
$\mathcal{E}_{x}(t)$, is the element of $\mathcal{T}$ given by left twist
deformation of $X$ along $x$ at the time $t$ starting at $X$. (Clearly,
$\mathcal{E}_{x}(t)$ also depends on $X$).

By [18, Proposition 3.5] and [15, Lemma 2.1] we have,

######  Lemma 2.4.

If $X\in\mathcal{T}$, and $x$ and $y$ are two $X$-geodesics such that $x$ is
simple, and $P$ is an $(x,y)$-intersection point then the function
$\phi_{P}(\mathcal{E}_{x}(t))$ is a strictly decreasing function of $t$.

######  Remark 2.5.

The aforementioned Proposition 3.5 of [18] is proved assuming that both $x$
and $y$ are simple geodesics. In the same work, it is stated that it holds
when $y$ is non-simple. An explicit proof of this fact can be found in [15,
Lemma 2.1].

We include the following result from [3, Theorem 7.38.6] and part of its proof
because both will be used later.

######  Theorem 2.6.

Let $a$ and $b$ be hyperbolic isometries of the hyperbolic plane, whose axes
intersect at a point $P$. Denote by $\beta$ the angle at $P$ of these axes in
the forward direction of $a$ and $b$. Then the product $a.b$ is hyperbolic and

$\cosh\left(\frac{t_{a.b}}{2}\right)=\cosh\left(\frac{t_{a}}{2}\right)\cosh\left(\frac{t_{b}}{2}\right)+\sinh\left(\frac{t_{b}}{2}\right)\sinh\left(\frac{t_{b}}{2}\right)\cos(\beta),$

where $t_{\alpha}$ denotes the translation length of $\alpha$.

###### Proof.

Denote by $Q$ the point on the axis of $a$ at distance $t_{a}/2$ of $P$ in the
positive direction of the axis of $a$ and by $R$ the point on the axis of $b$
at distance $t_{b}/2$ of $P$ in the negative direction of the axis of $b$. The
axis of $a.b$ is the geodesic containing the oriented line from $R$ to $Q$ and
the translation length of $a.b$ equals twice the distance between $R$ and $Q$.
(see Figure 3). The formula follows from the Cosine formula (see [3] for more
details).

Figure 3. Theorem 2.6

∎

For each closed curve $x$ on $\Sigma$, the length of the unique $X$-geodesic
in $\widetilde{x}$ is denoted by $\ell_{x}(X)$.

Figure 4. Lifts of ${(x*_{p}y)}_{0}$ and ${(x*_{p}y)}_{\infty}$.

Given two $X$-geodesics $x$ and $y$ and an $(x,y)$-intersection point $P$, a
lift of ${(x*_{p}y)}_{0}$ (respectively of ${(x*_{p}y)}_{\infty}$) to the
upper half plane ${\mathbb{H}}$ is a bi-infinite piecewise geodesic (see
Figure 4) consist of alternative geodesic segments of lift of the geodesics
$x$ and $y$ (denoted by $x_{*}$ and $y_{*}$ in Figure 4 respectively). The
geodesic segments of lift of the geodesics $x$ and $y$ intersect each other in
the lifts of the point $P$ (denoted by $P_{*}$ in Figure 4).

The next lemma follows directly from Theorem 2.6, and from Figure 4 by adding
an appropriate orientation to the geodesics $x$ and $y$.

######  Lemma 2.7.

If $x$ and $y$ are two closed $X$-geodesics and let $P$ be an
$(x,y)$-intersection point. Then we have

$\cosh\left(\frac{\ell_{(x*_{P}y)_{0}}}{2}\right)=\cosh\left(\frac{\ell_{x}}{2}\right)\cosh\left(\frac{\ell_{y}}{2}\right)-\sinh\left(\frac{\ell_{x}}{2}\right)\sinh\left(\frac{\ell_{y}}{2}\right)\cos(\phi_{P})$
$\cosh\left(\frac{\ell_{(x*_{P}y)_{\infty}}}{2}\right)=\cosh\left(\frac{\ell_{x}}{2}\right)\cosh\left(\frac{\ell_{y}}{2}\right)+\sinh\left(\frac{\ell_{x}}{2}\right)\sinh\left(\frac{\ell_{y}}{2}\right)\cos(\phi_{P}),$

where all lengths and angles are computed with respect to any metric $Y$ in
$\mathcal{T}$.

The next lemma states that, a $0$-term of the bracket of two curves, and an
$\infty$-term of the bracket of the same curves are always distinct when one
of them is simple. It is easier to state it in terms of geodesics (instead of
free homotopy classes of curves).

######  Lemma 2.8.

Let $X$ be a hyperbolic metric on $\Sigma$. If $x$ and $y$ are closed,
$X$-geodesics, such that $x$ is simple, and $P$ and $Q$ are two (not
necessarily distinct) $(x,y)$-intersection points then
$\widetilde{(x*_{P}y)_{0}}\neq\widetilde{({x}*_{Q}{y})_{\infty}}.$

###### Proof.

We argue by contradiction. If
$\widetilde{(x*_{P}y)_{0}}=\widetilde{({x}*_{Q}{y})_{\infty}}$ then
$\ell_{(x*_{P}y)_{0}}(Y)=\ell_{({x}*_{Q}{y})_{\infty}}(Y)$ for any
$Y\in\mathcal{T}$. By Lemma 2.7 we have

$\cos(\phi_{P}(Y))=-\cos(\phi_{Q}(Y)),$

which implies,

$\phi_{P}(Y)+\phi_{Q}(Y)=\pi.$ (1)

for all $Y\in\mathcal{T}$. On the other hand, as $x$ is simple, by Lemma 2.4,
by twisting the metric $X$ about the geodesic $x$, both terms on the right
side of Equation (1) strictly decrease. Since they add up to a constant, this
is not possible. Hence, the proof is complete. ∎

###  2.3. Proof of the counting intersection theorem

If $\widetilde{x}$ and $\widetilde{y}$ have disjoint representatives, the
result follows directly. Assume that $i(\widetilde{x},\widetilde{y})>0$. From
the definition of the bracket, it follows that

$\left[\widetilde{x},\widetilde{y}\right]=\sum\limits_{P\in x\cap
y}\widetilde{(x*_{P}y)}_{0}-\widetilde{(x*_{P}{y})}_{\infty}.$

Fix a metric $X\in\mathcal{T}$. In order to simplify the notation, assume that
$x$ and $y$ are $X$-geodesics. This implies that $x$ and $y$ intersect in
$i(\widetilde{x},\widetilde{y})$ points, the geometric intersection number of
the class.

Suppose that the number of terms of the bracket is strictly smaller than
$2.i(\widetilde{x},\widetilde{y})$. Hence, there exist two (not necessarily
distinct) (x,y)-intersection points $P$ and $Q$ such that a pair of terms
corresponding $P$ and $Q$ cancel.

The terms corresponding to $P$ are
$\widetilde{(x*_{P}y)_{0}}-\widetilde{({x}*_{P}{y})_{\infty}}$ and the terms
corresponding to $Q$ are
$\widetilde{(x*_{Q}y)}_{0}-\widetilde{({x}*_{Q}y)}_{\infty}.$

The assumption of cancellation implies that either
$\widetilde{(x*_{P}y)}_{0}=\widetilde{(x*_{Q}y)}_{\infty}$ or
$\widetilde{(x*_{P}y)}_{\infty}=\widetilde{(x*_{Q}y)}_{0}$ which is not
possible by Lemma 2.8. Thus, the proof is complete.

######  Remark 2.9.

In the case of the Goldman bracket of two directed curves, cancellation of two
terms (regardless whether they are simple or not) implies that the
corresponding directed angles are congruent, [15, Theorem 5.1]. (The directed
angle between two directed geodesics intersecting at a point $P$ is the angle
between the positive direction of both curves).

In the case of the TWG-bracket, cancellation of two terms implies that the
(undirected) angles are supplementary.

##  3\. TWG-Lie Bracket of powers of curves and proof of the Center Theorem

Let $X\in\mathcal{T}$. If $P$ is an $(x,y)$-intersection point of two
$X$-geodesics $x$ and $y$ then for each positive integer $m$, the geodesic
$x^{m}$ (that goes $m$ times around $x$) and $y$ also intersect at $P$.

######  Remark 3.1.

The angles at $P$ of $x$ and $y$ and of $x^{n}$ and $y$ are congruent (they
are the same angle). Thus, we can (and will) consider $P$ as
$(x^{m},y)$-intersection point and the angle $\phi_{P}(Y)$ will also denote
the angle at $P$ of the $Y$-geodesic homotopic to $x^{m}$ and the $Y$-geodesic
homotopic to $y$.

######  Proposition 3.2.

Let $X\in\mathcal{T}$ be hyperbolic metric on $\Sigma$. Let $x,y$ and $z$ be
three $X$-geodesics. Let $P$ and $Q$ be two $(x,y)$ and $(x,z)$-intersection
points respectively.

  1. (1)

If there exist two distinct positive values of $m$ such that
$\widetilde{(x^{m}*_{P}y)}_{0}=\widetilde{(x^{m}*_{Q}z)}_{0}$ (resp.
$\widetilde{(x^{m}*_{P}y)}_{\infty}=\widetilde{(x^{m}*_{Q}z)}_{\infty}$ ) then
$\ell_{y}(Y)=\ell_{z}(Y)$ and $\phi_{P}(Y)=\phi_{Q}(Y),$ for all
$Y\in\mathcal{T}.$

  2. (2)

If there exist two distinct positive values of $m$ such that
$\widetilde{(x^{m}*_{P}y)}_{0}=\widetilde{(x^{m}*_{Q}z)}_{\infty}$ then
$\ell_{y}(Y)=\ell_{z}(Y)$ and $\phi_{P}(Y)+\phi_{Q}(Y)=\pi$ for all
$Y\in\mathcal{T}$.

###### Proof.

We prove (1); the proof of (2) is analogous. From now on, we will fix a metric
$Y\in\mathcal{T}$. To simplify the notation, we will not write the dependence
on $Y$ (for instance, we will write $\cos(\phi_{P})$ instead of
$\cos(\phi_{P})(Y)$). We follow the notation indicated in Remark 3.1.

Since $\widetilde{(x^{m}*_{P}y)}_{0}=\widetilde{(x^{m}*_{Q}z)}_{0},$
$\ell_{(x^{m}*_{P}y)_{0}}=\ell_{(x^{m}*_{Q}z)_{0}}.$ By Lemma 2.7, we have

$\cosh(\frac{1}{2}\ell_{{x^{m}}})\cosh(\frac{1}{2}\ell_{{y}})-\sinh(\frac{1}{2}\ell_{{x^{m}}})\sinh(\frac{1}{2}\ell_{{y}})\cos(\phi_{P})=$
$\cosh(\frac{1}{2}\ell_{{x^{m}}})\cosh(\frac{1}{2}\ell_{{z}})-\sinh(\frac{1}{2}\ell_{{x^{m}}})\sinh(\frac{1}{2}\ell_{{z}})\cos(\phi_{Q}).$

This implies

$\coth(\frac{1}{2}\ell_{{x^{m}}})\\{\cosh(\frac{1}{2}\ell_{{y}})-\cosh(\frac{1}{2}\ell_{{z}})\\}=\sinh(\frac{1}{2}\ell_{{y}})\cos(\phi_{P})-\sinh(\frac{1}{2}\ell_{{z}})\cos(\phi_{Q})$

Note that that the right-hand side of the above equation does not depend on
$m$. Also, if $m_{1}$ and $m_{2}$ are distinct positive integers then
$\coth(\frac{1}{2}\ell_{x^{m_{1}}})\neq\coth(\frac{1}{2}\ell_{x^{m_{2}}}).$
This implies $\cosh(\frac{1}{2}\ell_{{y}})-\cosh(\frac{1}{2}\ell_{{z}})=0,$
and so, $\ell_{y}=\ell_{z}$. Hence, $\cos(\phi_{P})=\cos(\phi_{Q}),$ which
implies the equality of the corresponding angles, as desired. ∎

######  Lemma 3.3.

Let $X$ in $\mathcal{T}$ and let $x,y$ and $z$ be three $X$-geodesics in
$\Sigma$ such that $x$ is simple, $\ell_{y}(Y)=\ell_{z}(Y)$ for all
$Y\in\mathcal{T}$ and there exist an $(x,y)$-intersection point $P$ and a
$(x,z)$-intersection point $Q$ such that for some positive integer $m$,
$\widetilde{(x^{m}*_{P}y)}_{0}=\widetilde{(x^{m}*_{Q}z)}_{0}$. Then either
$y=z$ or there exist an $(x,y)$-intersection point $R$ such that
$\phi_{R}>\phi_{P}.$

###### Proof.

We prove the result for $m=1$. Combining Remark 3.1 with the equality
$\ell_{x^{m}}=m\ell_{x}$ the proof for $m>1$ follows by a similar argument.

Since $\widetilde{(x*_{P}y)_{0}}=\widetilde{(x*_{Q}z)_{0}}$, there exists two
lifts to the universal cover of the surface, the hyperbolic plane
${\mathbb{H}}$, one of the piecewise geodesics $(x*_{P}y)_{0}$ and the other
of $(x*_{Q}z)_{0}$ with the same endpoints. Denote these two lifts by $C$ and
$D$ respectively, and by $L$ the geodesic line joining their common endpoints.
The two piecewise geodesics $C$ and $D$ zigzag about the line $L$. The zigzag
curve $C$ (resp. $D$) is composed of alternating segments of lifts of $x$ of
length $\ell_{x}$, and lifts of $y$ (resp. of $z$) of length $\ell_{y}$. In
Figure 5, “laps” of lifts of $x$ are represented in blue, “laps” of lifts of
$y$ in green and “laps” of lifts of $z$ in red. The line $L$ intersects each
of these lap segments in their midpoints. (See Theorem 2.6 and its proof).

Consider a segment $S$ of the zigzag curve $C$, which is a lift of $x$. Denote
the intersection point of $S$ and $L$ by $U$. Choose an endpoint of $S$ and
denote it by $P_{1}$. Denote by $V$ the intersection of $L$ with the other
segment of $C$ with endpoint $P_{1}$ (this last segment is a lift of $y$).
These three points determine a triangle $UVP_{1}$

Consider the triangle $U^{\prime}V^{\prime}Q_{1}$, analogous to $UVP_{1}$, but
with sides included in the zigzag curve $D$.

Note that the length of both segments, $UP_{1}$ and $UQ_{1}$ is $\ell_{x}/2$.
Also the length of $VP_{1}$ and $V^{\prime}Q_{1}$ is $\ell_{y}/2$. By Lemma
2.7, the length of $UV$ and $U^{\prime}V^{\prime}$ is half the length of the
geodesic in $\widetilde{(x*_{P}y)_{0}}$. Therefore, these two triangles
$UVP_{1}$ and $U^{\prime}V^{\prime}Q_{1}$ are congruent. Hence, there is an
isometry mapping one triangle to the other. If this isometry is orientation
reversing, then it maps $U$ to $U$, $V$ to $V^{\prime}$ and $P_{1}$ to
$Q_{1}$. This implies that $\phi_{P}(X)+\phi_{Q}(X)=\pi$, see Figure 5, a.

If we perturb the metric $X$ slightly, we can repeat the above argument, and
obtain that $\phi_{P}(Y)+\phi_{Q}(Y)=\pi$, for all $Y$ in a neighborhood of
$X$. (The orientation reversing isometry for the corresponding $Y$-geodesics
must exist by continuity). Since $x$ is simple, this is not possible by Lemma
2.4. Thus, there is an orientation preserving isometry mapping $U$ to $U$, $V$
to $V^{\prime}$ and $P_{1}$ to $Q_{1}$. Now, there are two possibilities:
either the midpoint of a lap of a lift of $x$ is also a midpoint of lap of a
lift or $y$ (Figure 5, right) or not (Figure 5, middle.)

Figure 5. Zigzags

If $C=D$, then $y=z$ and the proof is complete. Hence, we can assume $C\neq
D$. There are then two cases left, depicted in Figure 5, b. and c. In the case
illustrated in Figure 5, b., a segment lifting of $z$ intersects the interior
of the triangle $UVP_{1}$, and determines a triangle $WVR$ as in the figure.
Since the area of $WVR$ is smaller than that of $UVP_{1}$, and two of the
angles of of $WVR$ are congruent to two of the angles of $UVP_{1}$, the angle
at $P_{1}$, is smaller than the angle at $R$, so the proof of this case is
complete.

In the case illustrated in Figure 5, c.,
$\phi_{P}+\alpha+\beta<\pi=\phi_{R}+\alpha+\beta$, and $\phi_{P}<\phi_{R}$, as
desired. ∎

######  Proposition 3.4.

Let $X\in\mathcal{T}$ be a metric on $\Sigma$ and $x,y,z$ be three pairwise
distinct $X$-geodesics such that $x$ is simple and let $P$ and $Q$ be $(x,y)$
and $(x,z)$-intersection points respectively. The following holds.

  1. (1)

The equality
$\widetilde{(x^{m}*_{P}y)}_{0}=\widetilde{(x^{m}*_{Q}z)}_{\infty}$ holds for
at most one positive value of $m$.

  2. (2)

Either the equality
$\widetilde{(x^{m}*_{P}y)}_{0}=\widetilde{(x^{m}*_{Q}z)}_{0}$ (resp.
$\widetilde{(x^{m}*_{P}y)}_{\infty}=\widetilde{(x^{m}*_{Q}z)}_{\infty}$) holds
for at most one positive value of $m$ or there exist an $(x,y)$-intersection
point $R$ such that $\phi_{P}(X)<\phi_{R}(X).$

###### Proof.

Suppose $\widetilde{(x^{m}*_{P}y)}_{0}=\widetilde{(x^{m}*_{Q}z)}_{\infty}$ for
two distinct values of $m$. By Proposition 3.2(2),
$\phi_{P}(Y)+\phi_{Q}(Y)=\pi,$ for all $Y\in\mathcal{T}.$ By Lemma 2.4 this is
not possible. Thus, (1) is proved.

If $\widetilde{(x^{m}*_{P}y)}_{0}=\widetilde{(x^{m}*_{Q}z)}_{0},$ for more
than two values of $m$ by Proposition 3.2(1), we have that
$\ell_{y}(Y)=\ell_{z}(Y)$ and $\phi_{P}(Y)=\phi_{Q}(Y)$ for all
$Y\in\mathcal{T}$.

Fix $X\in\mathcal{T}$, any $m$ and consider a geodesic lift $A$ of the
geodesic in the free homotopy class of
$\widetilde{(x^{m}*_{P}y)}_{0}=\widetilde{(x^{m}*_{Q}z)}_{0}$.

The result follows them by Lemma 3.3.

The proof for the case
$\widetilde{(x^{m}*_{P}y)}_{\infty}=\widetilde{(x^{m}*_{Q}z)}_{\infty}$ is
similar. Hence, result is proved. ∎

######  Lemma 3.5.

Let $\widetilde{x},\widetilde{y_{1}},\dots,\widetilde{y_{k}}$ be pairwise
distinct free homotopy classes of closed curves such that $\widetilde{x}$
contains a simple representative. Let
$\widetilde{y}=\sum_{i=1}^{k}c_{i}\widetilde{y_{i}}$ where the coefficients
$c_{1},c_{2},\dots,c_{k}$ are in the ring $\mathbb{K}$. Then either
$i(x,y_{i})=0$ for all $i\in\\{1,\ldots,k\\}$ or there exists a positive
integer $m_{0}$ such that $[\widetilde{x}^{m},\widetilde{y}]\neq 0$ for all
$m\geq m_{0}$ .

###### Proof.

From the definition of the bracket, for any $m\in\mathbb{N}$,

$\displaystyle[\widetilde{x}^{m},\widetilde{y}]$
$\displaystyle=\sum_{i=1}^{k}c_{i}[\widetilde{x}^{m},\widetilde{y}_{i}]$
$\displaystyle=m\sum_{i=1}^{k}c_{i}\sum\limits_{P\in x\cap
y_{i}}\widetilde{(x^{m}*_{P}y_{i})}_{0}-\widetilde{(x^{m}*_{P}{y_{i}})}_{\infty}.$

Fix a metric $X$ and assume that $x$, $y_{1}\dots,y_{k}$ are $X$-geodesics.
For each $m$, the sum $\sum_{i=1}^{k}c_{i}\sum\limits_{P\in x\cap
y_{i}}\widetilde{(x^{m}*_{P}y_{i})}_{0}-\widetilde{(x^{m}*_{P}{y_{i}})}_{\infty}$
has
$I=2(i(\widetilde{x},\widetilde{y_{1}})+i(\widetilde{x},\widetilde{y_{2}})+\cdots+i(\widetilde{x},\widetilde{y_{k}}))$
terms, before performing possible cancellations.

Choose a metric $X$ in $\mathcal{T}$. Consider $P$, one of the intersection
points of the $X$-geodesic in $x$ and $\bigcup\limits_{1\leq i\leq k}y_{i}$
and choose one of the terms of the bracket associated with $P$, $0$ or
$\infty$. For simplicity, assume that the chosen term is
$(\widetilde{x*_{P}y_{1}})_{0}$.

If for all $1\leq m\leq I+1$, we have that
$[\widetilde{x}^{m},\widetilde{y}]=0$ then there exist
$m_{1},m_{2}\in\\{1,2,\dots,I+1\\}$, and an $X$-geodesic $y_{i}$, such that
one of the following holds:

  1. (a).

$(\widetilde{x^{m}*_{P}y_{1}})_{0}=(\widetilde{x^{m}*_{Q}y_{i}})_{0}$ for
$m=m_{1}$ and $m=m_{2}$.

  2. (b).

$(\widetilde{x^{m}*_{P}y_{1}})_{0}=(\widetilde{x^{m}*_{Q}y_{i}})_{\infty}$ for
$m=m_{1}$ and $m=m_{2}$.

By Proposition 3.4(1), (a) is not possible. Hence (b) holds and by Proposition
3.4(2), there exists a point $R$ in $x\cap y_{1}$, such that the angle $\phi$
at $P$ is strictly smaller than the angle $\phi$ at $R$ (in symbols,
$\phi_{P}<\phi_{R}$). This is not possible because $P$ was chosen arbitrarily,
and so the proof is complete. ∎

We will make use of the following well known result. (This result is usually
stated for finite type surfaces but it can be generalized to all Riemann
surfaces using the fact that every closed geodesic is included in a compact
subsurface).

######  Lemma 3.6.

If $\Sigma$ is an orientable surface and $y$ is a closed curve on $\Sigma$
such that $i(x,y)=0$ for every simple closed curve $x$. Then $y$ is either
homotopically trivial or homotopic to a boundary curve or homotopic to a
puncture.

###  3.1. Proof of the Center Theorem

Suppose that $\widetilde{y}=\sum_{i=1}^{k}c_{i}\widetilde{y_{i}}$ belongs to
the center, where the free homotopy classes of
$\widetilde{y}_{1},\widetilde{y}_{2},\dots,\widetilde{y}_{k}$ are pairwise
distinct and $c_{i}\in\mathbb{K}$ for $i\in\\{1,2,\ldots,k\\}$. Let $x$ be any
simple closed curve. By definition of center,
$[\widetilde{x}^{n},\widetilde{y}]=0$ for all positive integers $n$. Therefore
Lemma 3.5 implies that $i(x,y_{i})=0$ for all $i\in\\{1,2,\ldots k\\}$. Hence
Lemma 3.6 implies that each $\widetilde{y}_{i}$ is either homotopically
trivial or homotopic to a boundary curve or homotopic to a puncture, which
completes the proof.

##  4\. Universal enveloping algebra and symmetric algebra of TWG-Lie algebra

Let $\mathcal{U}(\mathbb{K}\widetilde{\pi})$ be the universal enveloping
algebra and $\mathcal{S}(\mathbb{K}\widetilde{\pi})$ be the symmetric algebra
of $K(\widetilde{\pi})$. For definition and basic properties of these objects
see [1], [13].

$\mathcal{U}(\mathbb{K}\widetilde{\pi})$ has a natural Poisson algebra
structure with the commutator being the Lie bracket. We extend the Lie bracket
of $\mathbb{K}\widetilde{\pi}$ to $\mathcal{S}(\mathbb{K}(\widetilde{\pi}))$
using Leibniz rule. This makes $\mathbb{K}(\widetilde{\pi})$ a Poisson
algebra.

The Poisson center of a Poisson algebra $(A,\\{,\\})$ is the subalgebra
consists of elements $y\in A$ such that $\\{x,y\\}=0$ for all $x\in A$. In
this section we discuss the Poisson center of the Poisson algebras
$\mathcal{U}(\mathbb{K}\widetilde{\pi})$ and
$\mathcal{S}(\mathbb{K}\widetilde{\pi})$.

There are canonical maps from $\mathbb{K}\widetilde{\pi}$ to
$\mathcal{U}(\mathbb{K}\widetilde{\pi})$ and
$\mathcal{S}(\mathbb{K}\widetilde{\pi})$. To simplify notation, we denote an
element and its image under these maps by the same notation. We also denote
the product of two elements in $\mathcal{U}(\mathbb{K}\widetilde{\pi})$ and
$\mathcal{S}(\mathbb{K}\widetilde{\pi})$ simply by juxtaposing the elements.

Let us recall the Poincare-Birkhoff-Witt theorem for
$\mathbb{K}\widetilde{\pi}$(see [1], [13, Theorem 3.2]).

######  Theorem 4.1.

Let $\leq$ be a fixed total order on $\widetilde{\pi}$. Consider the set

$S=\\{\widetilde{x}_{1}\widetilde{x}_{2}\cdots\widetilde{x}_{n}:n>0,\widetilde{x}_{i}\in\widetilde{\pi},i\in\\{1,2,\ldots,n\\},\widetilde{x}_{1}\leq\widetilde{x}_{2}\leq\cdots\leq\widetilde{x}_{n}\\}.$

Both $\mathcal{U}(\mathbb{K}\widetilde{\pi})$ and
$\mathcal{S}(\mathbb{K}\widetilde{\pi})$ are freely generated by $S$ as a
$\mathbb{K}$ module. Moreover the natural maps from $\widetilde{\pi}$ into
$\mathcal{U}(\mathbb{K}\widetilde{\pi})$ and
$\mathcal{S}(\mathbb{K}\widetilde{\pi})$ are injective Lie algebra
homomorphisms.

######  Theorem 4.2.

The Poisson center of the Poisson algebras
$\mathcal{U}(\mathbb{K}\widetilde{\pi})$ and
$\mathcal{S}(\mathbb{K}\widetilde{\pi})$ are generated by scalars
$\mathbb{K}$, the free homotopy class of constant curve and the curves
homotopic to boundary and punctures.

###### Proof.

We compute the center $\mathcal{U}(\mathbb{K}\widetilde{\pi})$. For
$\mathcal{S}(\mathbb{K}\widetilde{\pi})$, the proof is exactly the same. Let
$Z$ be an element of the center of $\mathcal{U}(\mathbb{K}\widetilde{\pi})$.
Then by Theorem 4.1,

$Z=\sum_{i=1}^{n}C_{i}\;\widetilde{x}_{i_{1}}\widetilde{x}_{i_{2}}\cdots\widetilde{x}_{i_{k}}$

where
$\widetilde{x}_{i_{1}}\leq\widetilde{x}_{i_{2}}\leq\cdots\leq\widetilde{x}_{i_{k}}$
for all $i\in\\{1,2,\ldots,n\\}$. For any
$\widetilde{x}\in\mathbb{K}\widetilde{\pi}$ we have

$\displaystyle 0=[\widetilde{x}^{n},Z]$
$\displaystyle=\sum_{i=1}^{n}C_{i}\;[x^{n},\widetilde{x}_{i_{1}}\widetilde{x}_{i_{2}}\cdots\widetilde{x}_{i_{k}}]$
$\displaystyle=\sum_{i=1}^{n}C_{i}\;\sum_{j=1}^{k}n\Bigg{\\{}\sum_{p_{i_{j}}\in\widetilde{x}\cap\widetilde{x}_{i_{j}}}\Big{(}\widetilde{x}_{i_{1}}\cdots\widetilde{x}_{i_{j}-1}(\widetilde{x}^{n}*_{P_{i_{j}}}\widetilde{x}_{i_{j}})_{0}\widetilde{x}_{i_{j}+1}\cdots\widetilde{x}_{i_{k}}$
$\displaystyle-\widetilde{x}_{i_{1}}\cdots\widetilde{x}_{i_{j}-1}(\widetilde{x}^{n}*_{P_{i_{j}}}\widetilde{x}_{i_{j}})_{\infty}\widetilde{x}_{i_{j}+1}\cdots\widetilde{x}_{i_{k}}\Big{)}\Bigg{\\}}$

Fix a metric $X\in\mathcal{T}$ and choose ${x}$ to be any simple $X$-geodesic.
Let $x_{i_{j}}$ be the $X$-geodesic in the free homotopy class of
$\widetilde{x}_{i_{j}}$.

Fix any $\widetilde{x}_{i_{j}}$ and consider $P\in{x}\cap{x}_{i_{j}}$ such
that $\phi_{P}(x,x_{i_{j}})\geq\phi_{R}(x,x_{i_{j}})$ for all $R\in x\cap
x_{i_{j}}$. As $[\widetilde{x}^{n},Z]=0$ for all positive integer $n$, from
the above expression and Theorem 4.1, there exists
$\widetilde{x}_{k_{l}}\neq\widetilde{x}_{i_{j}}$ such that for infinitely many
positive integer $n$, one of the following is true.

  * •

$(\widetilde{x}^{n}*_{P}\widetilde{x}_{i_{j}})_{0}=(\widetilde{x}^{n}*_{Q}\widetilde{x}_{k_{l}})_{0}$
for some $Q\in\widetilde{x}_{i_{j}}\cap\widetilde{x}_{k_{l}}$. This is
impossible by Proposition 3.4.

  * •

$(\widetilde{x}^{n}*_{P}\widetilde{x}_{i_{j}})_{0}=(\widetilde{x}^{n}*_{Q}\widetilde{x}_{k_{l}})_{\infty}$
for some $Q\in\widetilde{x}_{i_{j}}\cap\widetilde{x}_{k_{l}}$. This is
impossible by Proposition 3.4.

  * •

$(\widetilde{x}^{n}*_{P}\widetilde{x}_{i_{j}})_{0}=(\widetilde{x}_{k_{l}})$.
This is impossible by Lemma 2.7.

Therefore each $\widetilde{x}_{i_{j}}$ is disjoint from $\widetilde{x}$. As
$\widetilde{x}$ is an arbitrary simple closed geodesic, each
$\widetilde{x}_{i_{j}}$ is disjoint from every simple closed geodesic on the
surface. Hence by Lemma 3.6 each $\widetilde{x}_{i_{j}}$ is either a constant
loop or a loop homotopic to a puncture or a loop homotopic to a boundary
component. ∎

##  5\. Examples of the Goldman and TWG brackets

Recall that, on a path connected space, the set of free homotopy classes of
closed directed curves is in one to one correspondence with conjugacy classes
of the fundamental group. Thus, the set of conjugacy classes of the
fundamental group minus the conjugacy class of the trivial loop is in two-to-
one correspondence with the set of free homotopy classes of oriented closed
curves minus the class of the trivial loop. Recall also that the fundamental
group of a surface with boundary is in one to one correspondence with the set
of reduced words in a fixed minimal set of generators. Hence, the set of free
homotopy classes of closed directed curves on a surface with boundary is in
one to one correspondence with the set of cyclic, reduced words on a fixed
minimal set of generators. (The empty word is also a reduced word in this
discussion)

We will use the above correspondences in this section, where we give examples
of the Goldman and TWG bracket on certain surfaces with boundary, first by
choosing a set of minimal generators of the fundamental group, and second,
denoting each class of curves by a cyclic word in these generators and their
inverses. (Given the two-to-one correspondence mentioned above, there are two
possible cyclic words denoting a class of undirected curves). To simplify the
notation, the inverse of a generator $x$ will be denoted by $X$. Also,
$\widetilde{w}$ will denote the free homotopy class of a representative of $w$
after removing the orientation and the basepoint.

Figure 6. A choice of generators of the pair of pants (left) and the punctured
torus (right)

######  Example 5.1.

Consider the fundamental group of the triply punctured sphere or pair of pants
with generators given by the classes $a$ and $b$ of two curves parallel to two
of the boundary components, oriented so that $ab$ goes around the third
boundary component (see Figure 6, left).

Figure 7. Example $[\widetilde{aab},\widetilde{aB}]$: Representatives of
$\widetilde{aab}$ and $\widetilde{aB}$ (left) and computation of the bracket
(right)

$[\widetilde{aab},\widetilde{aB}]=\widetilde{baaBa}-\widetilde{Baaba},$

Example 5.1 shows the need of the hypothesis of one of the curves being simple
in the Intersection Counting Theorem since
$i(\widetilde{aab},\widetilde{aB})=2$ but the number of terms of the bracket
is less than twice the intersection number, which is $4$.

######  Example 5.2.

In the same setting as Example 5.1, it is not hard to check that the Goldman
bracket of the conjugacy classes of the directed curves $aaB$ and $aB$ and the
two curves have intersection number equal to $2$.

######  Example 5.3.

If we consider the punctured torus with fundamental group with standard
generators labeled $a$ and $b$ then
$[\widetilde{abAb},\widetilde{aB}]=\widetilde{aBBB}-\widetilde{ABaBAb}+\widetilde{AB}-\widetilde{aBABaB}.$

Figure 8. Example $[\widetilde{abAb},\widetilde{aB}]$: Representatives of
$\widetilde{abAb}$ and $\widetilde{aB}$ (left) and computation of the bracket
(right)

$[\widetilde{aab},\widetilde{aB}]=\widetilde{baaBa}-\widetilde{Baaba},$

$[\widetilde{aaB},\widetilde{aB}]=\widetilde{aaBAb}-\widetilde{aabAB}.$

Example 5.3 illustrates Intersection Counting Theorem. In this case,
$i(\widetilde{aB},\widetilde{aab})=2$ and the number of terms of the bracket
$[\widetilde{aab},\widetilde{ab}]$ is $4$.

######  Theorem (Computational).

Consider two classes of curves $\widetilde{x}$ and $\widetilde{y}$, If one of
the following holds

  * •

The word length of $\widetilde{x}$ and $\widetilde{y}$ is less than or equal
to $12$ and $x$ and $y$ are in the punctured torus.

  * •

The word length of $\widetilde{x}$ and $\widetilde{y}$ is less than or equal
to $11$ and $x$ and $y$ are in the pair of pants.

  * •

The word length of $\widetilde{x}$ and $\widetilde{y}$ is less than or equal
to $8$ and $x$ and $y$ are in $aAbBcC$ the four holed sphere.

  * •

The word length of $\widetilde{x}$ and $\widetilde{y}$ is less than or equal
to $7$ and $x$ and $y$ are in the punctured genus two surface with surface
word $abABcdCD$.

and the bracket of $\widetilde{x}$ and $\widetilde{y}$ is zero then
$\widetilde{x}$ and $\widetilde{y}$ have disjoint representatives. In symbols,
if $[\widetilde{x},\widetilde{y}]=0$ then $i(\widetilde{x},\widetilde{y})=0.$

The previous Computational Theorem lead us to the following conjecture.

######  Conjecture.

If the TWG-Lie bracket of two classes of undirected curves is zero, then the
classes have disjoint representatives.

##  6\. Proof of the Jacobi identity for the TWG-bracket

A pair of points $(P,Q),P\in x\cap y,Q\in y\cap z$ determines four terms of
the double bracket $[[\widetilde{x},\widetilde{y}],\widetilde{z}]$, each of
these four types represented in the upper row of Figure 9. Repeating this
reasoning for the other two brackets,
$[[\widetilde{y},\widetilde{z}],\widetilde{x}]$ and
$[[\widetilde{z},\widetilde{x}],\widetilde{y}]$, it is not hard to check the
Jacobi identity.

$[[\widetilde{x},\widetilde{y}],\widetilde{z}]+[[\widetilde{y},\widetilde{z}],\widetilde{x}]+[[\widetilde{z},\widetilde{x}],\widetilde{y}]=0.$

Figure 9. Representation of the double brackets
$[[\widetilde{x},\widetilde{y}],\widetilde{z}]$,
$[[\widetilde{y},\widetilde{z}],\widetilde{x}]$, and
$[[\widetilde{z},\widetilde{x}],\widetilde{y}]$

##  7\. Goldman’s definition of the TWG-bracket

In this section we review the definition of the Goldman Lie algebra on
directed curves, Goldman’s definition of the TWG-Lie algebra of undirected
curves and prove that there is an isomorphism between Goldman’s and the
definition of the TWG-Lie algebra we gave in the introduction.

Denote by $\pi$ the set of free homotopy classes of directed closed curves on
$\Sigma$, by $\langle\alpha\rangle$ the free homotopy class of a directed
closed curve $\alpha$, and by $\mathbb{K}\pi$ the free module spanned by
$\pi$.

The _Goldman Lie bracket on $\mathbb{K}\pi$_ is the linear extension to
$\mathbb{K}\pi$ of the bracket of two free homotopy classes
$\langle\alpha\rangle$ and $\langle\beta\rangle$ defined by

$[\langle\alpha\rangle,\langle\beta\rangle]=\sum_{P\in\alpha\cap\beta}\varepsilon_{P}\langle\alpha*_{P}\beta\rangle.$

Here, the representatives $\alpha$ and $\beta$ are chosen so that they
intersect transversely in a set of double points $\alpha\cap\beta$,
$\varepsilon_{P}$ denotes the sign of the intersection between $\alpha$ and
$\beta$ at an intersection point $P$, and $\alpha*_{P}\beta$ denotes the loop
product of ${\alpha}$ and ${\beta}$ at $P$.

Goldman [11] proved that this bracket is well defined, skew-symmetric and
satisfies the Jacobi identity on $\mathbb{K}\pi$. In other words,
$\mathbb{K}\pi$ is a Lie algebra.

There is a natural involution ${\iota}\colon{\pi}\longrightarrow{\pi}$ defined
by $\iota(\langle\alpha\rangle)=\langle\bar{\alpha}\rangle$ where
$\bar{\alpha}$ denotes the curve $\alpha$ with opposite orientation. By
extending $\iota$ linearly to $\mathbb{K}\pi$ we obtain a $\mathbb{K}$ -linear
involution ${\iota}\colon{\mathbb{K}\pi}\longrightarrow{\mathbb{K}\pi}$. The
invariant subspace of $\iota$, denoted by $S$ is a Lie subalgebra of
$\mathbb{K}\pi$ [11, Subsection 5.12].

Now we prove that the subalgebra $S$ is isomorphic to the TWG-Lie algebra
$\mathbb{K}\widetilde{\pi}$.

First observe that $S$ is generated by the elements of the form
$\langle\alpha\rangle+\langle\bar{\alpha}\rangle$. A straightforward
computation in $\mathbb{K}\widetilde{\pi}$ shows that

$[\langle\alpha\rangle+\langle\bar{\alpha}\rangle,\langle\beta\rangle+\langle\bar{\beta}\rangle]=[\langle\alpha\rangle,\langle\beta\rangle]+\overline{[\langle\alpha\rangle,\langle\beta\rangle]}+[\langle\alpha\rangle,\langle\bar{\beta}\rangle]+\overline{[\langle\alpha\rangle,\langle\bar{\beta}\rangle]}$
(2)

where the “change direction” operator $\overline{\leavevmode\nobreak\
\cdot\leavevmode\nobreak\ }$ is extended to $\mathbb{K}\widetilde{\pi}$ by
linearity.

Next define a map from $S$ to $\mathbb{K}\widetilde{\pi}$, by sending each
element of $S$ of the form $\langle\alpha\rangle+\langle\bar{\alpha}\rangle$
(that is, each element of a the geometric basis of $S$) to the undirected free
homotopy class $u_{\langle\alpha\rangle}$ defined by “forgetting” the
direction of $\alpha,$ and considering the free homotopy class. Extend $u$ to
$S$ by linearity.

Figure 10. Sign of the elements.

Observe that (Figure 10)

$\varepsilon_{P}(\alpha,\beta)=\varepsilon_{P}(\bar{\alpha},\bar{\beta})=-\varepsilon_{P}(\alpha,\bar{\beta})=-\varepsilon_{P}(\bar{\alpha},\beta).$

Also, if $\varepsilon_{P}(\alpha,\beta)=1$ then

$u_{\langle\alpha*_{P}\beta\rangle}=\widetilde{(\alpha*_{P}\beta)}_{0}\,\,\text{and}\,\,u_{\langle\alpha*_{P}\bar{\beta}\rangle}=\widetilde{(\alpha*_{P}\beta)}_{\infty}.$

and if $\varepsilon_{P}(\alpha,\beta)=-1$ then

$u_{\langle\alpha*_{P}\beta\rangle}=\widetilde{(\alpha*_{P}\beta)}_{\infty}\,\,\text{and}\,\,u_{\langle\alpha*_{P}\bar{\beta}\rangle}=\widetilde{(\alpha*_{P}\beta)}_{0}.$

Therefore computing the bracket in $\mathbb{K}\widetilde{\pi}$, we observe

$[u_{\langle\alpha\rangle},u_{\langle\beta\rangle}]=\sum_{P\in\alpha\cap\beta,\varepsilon_{P}=1}\left(u_{\langle\alpha*_{P}\beta\rangle}-u_{\langle\alpha*_{P}\bar{\beta}\rangle}\right)-\sum_{P\in\alpha\cap\beta,\varepsilon_{P}-1}\left(u_{\langle\alpha*_{P}\bar{\beta}\rangle}-u_{\langle\alpha*_{P}\beta\rangle}\right)$
$=\sum_{P\in\alpha\cap\beta}\varepsilon_{P}\left(u_{\langle\alpha*_{P}\beta\rangle}+u_{\langle\alpha*_{P}\bar{\beta}\rangle}\right)=u_{[\langle\alpha\rangle,\langle\beta\rangle]}+u_{[\langle\alpha\rangle,\langle\bar{\beta}\rangle]}$

On the other hand, by Equation (2) by applying $u$ to the bracket
$[\langle\alpha\rangle+\langle\bar{\alpha}\rangle,\langle\beta\rangle+\langle\bar{\beta}\rangle]$
we obtain
$u_{[\langle\alpha\rangle,\langle\beta\rangle]}+u_{[\langle\alpha\rangle,\langle\bar{\beta}\rangle]}.$
This shows that $u$ is a Lie algebra isomorphism, as desired.

##  8\. Proof of the Canonical Decomposition Theorem

We now prove the Canonical Decomposition Theorem. Consider two undirected
classes of curves $\alpha$ and $\beta$, not null-homotopic and not parallel to
a boundary component in the surface $\Sigma$. Fix a hyperbolic metric $X$ on a
surface, which is homotopy equivalent to $\Sigma$, so that all boundary
components are punctures. Let $x$ and $y$ be $X$-geodesic representatives of
$\alpha$ and $\beta$. By Theorem 2.6, both $(x*_{P}y)_{0}$ and
$(x*_{P}y)_{\infty}$ are hyperbolic and have positive length. Thus, they are
not parallel to boundary components and not null-homotopic.

The proof in the case of the Goldman Lie algebra follows similarly.

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