# Block mapping class groups and their finiteness properties

J. Aramayona  Javier Aramayona: Instituto de Ciencias Matemáticas (ICMAT).
Nicolás Cabrera, 13–15. 28049, Madrid, Spain<EMAIL_ADDRESS>,  J.
Aroca  Julio Aroca: Instituto de Ciencias Matemáticas (ICMAT). Nicolás
Cabrera, 13–15. 28049, Madrid, Spain<EMAIL_ADDRESS>,  M. Cumplido
María Cumplido: Departmento de Álgebra, Facultad de Matemáticas, Universidad
de Sevilla Calle Tarfia s/n 41012, Seville, Spain<EMAIL_ADDRESS>,  R.
Skipper  Rachel Skipper: Department of Mathemtics, The Ohio State University,
231 W 18th Ave, Columbus, OH 43210, USA<EMAIL_ADDRESS>and  X. Wu
Xiaolei Wu: Shanghai Center for Mathematical Sciences, Jiangwan Campus, Fudan
University, No.2005 Songhu Road,Shanghai, 200438 P.R. China
<EMAIL_ADDRESS>
###### Abstract.

Given $g\in\mathbb{N}\cup\\{0,\infty\\}$, let $\Sigma_{g}$ denote the closed
surface of genus $g$ with a Cantor set removed, if $g<\infty$; or the blooming
Cantor tree, when $g=\infty$. We construct a family $\mathfrak{B}(H)$ of
subgroups of $\operatorname{Map}(\Sigma_{g})$ whose elements preserve a block
decomposition of $\Sigma_{g}$, and eventually like act like an element of $H$,
where $H$ is a prescribed subgroup of the mapping class group of the block.
The group $\mathfrak{B}(H)$ surjects onto an appropriate symmetric Thompson
group of Farley–Hughes [FH15]; in particular, it answers [AV20, Question 5.37]
positively.

Our main result asserts that $\mathfrak{B}(H)$ is of type $F_{n}$ if and only
if $H$ is. As a consequence, for every $g\in\mathbb{N}\cup\\{0,\infty\\}$ and
every $n\geq 1$, we construct a subgroup $G<\operatorname{Map}(\Sigma_{g})$
that is of type $F_{n}$ but not of type $F_{n+1}$, and which contains the
mapping class group of every compact surface of genus $\leq g$ and with non-
empty boundary.

J.A. was supported by grant CEX2019-000904-S funded by MCIN/AEI/
10.13039/501100011033 and by grant PGC2018-101179-B-I00. M.C. was supported by
the Spanish grants P20_01109 (financed by Junta de Andalucía) and
PID2020-117971GB-C21 (financed by Ministerio de Ciencia e Innovación, Spain),
the Research Program “Braids” of ICERM (Providence, RI), and a Ramón y Cajal
Grant 2021 (financed by Ministerio de Ciencia e Innovación, Spain) . R.S. was
partially supported by NSF grant DMS–2005297. X.W. was partially supported by
the Start-up Grant of Fudan University.

##  1\. Introduction

Asymptotic mapping class groups of surfaces offer a fascinating bridge between
mapping class groups of compact surfaces and Higman-Thompson groups [Bri07,
Deh06, FK04, FK08, FK09, AF21, GLU20, SW21a, Ara+21]. Originally introduced by
Funar–Kapoudjian [FK04, FK09] as discrete versions of the diffeomorphism group
of the circle, they are strongly related to other classical families of groups
that serve as “artinizations” of Higman-Thompson groups, such as the braided
Thompson groups of Brin [Bri07] and Dehornoy [Deh06]; see [FKS13] for an
account on this topic.

A substantial piece of motivation for asymptotic mapping class groups is that
they are subgroups of mapping class groups of infinite-type surfaces, now
commonly known as big mapping class groups. In this direction, a recent result
of Funar–Neretin [FN18] asserts that, the subgroup of the mapping class group
of a closed surface minus a Cantor set whose elements can be smoothly extended
to the closed surface coincides with a certain asymptotic mapping class group
(the group of half-twists of [FNg]). Moreover, by a result of Funar and the
first-named author, this group is in fact dense in the mapping class group of
the underlying surface.

Work of many authors shows that asymptotic mapping class groups –as well as
some of their Thompson-like relatives– tend to enjoy strong finiteness
properties; more concretely, they are often of type $F_{\infty}$ [BG84, FH15,
Flu+13, Bux+16, GLU20, SW21, Ara+21].

In this paper we introduce a more general family of groups, which this time
surject onto the symmetric Higman-Thompson groups of Farley–Hughes [FH15]
(more generally the Neretin groups [Ner92]), thus providing an answer to
[AV20, Question 5.37]. Moreover, we will prove that they realize all the
possible types of finiteness properties. We now offer an informal definition
of our groups sufficient to understand our main result, and postpone all
details until Section 3.

Let $O$ and $Y$ be compact, connected, orientable surfaces, where $Y$ is
further assumed to be closed. For each $d\geq 2$, write $Y^{d}$ for the
surface that results from removing $d+1$ open disks from $Y$. From this data,
we construct a non-compact surface ${\mathcal{C}}_{d}(O,Y)$ by gluing copies
of $Y^{d}$ to $O$ in a tree-like manner; see Figure 1. Following [Ara+21], the
surface ${\mathcal{C}}_{d}(O,Y)$ is called the Cantor surface determined by
$d$, $O$ and $Y$. Each of standard copies of $Y^{d}$ used to construct
${\mathcal{C}}_{d}(O,Y)$ is called a block. We will be interested in the group
$\mathfrak{B}_{d}(H)$ whose elements are isotopy classes of homeomorphisms of
${\mathcal{C}}_{d}(O,Y)$ which eventually send blocks to blocks and, on each
block in a connected component, they act like an element of a fixed subgroup
$H$ of the mapping class group of the block; see Section 3 for details. In the
case when $H$ is trivial, the group $\mathfrak{B}_{d}(H)$ is the asymptotic
mapping class group of [Ara+21].

As we will see in Proposition 4.2, there is a surjective homeomorphism from
$\mathfrak{B}_{d}(H)$ onto an appropriate symmetric Higman-Thompson group,
which simply records the action of elements of $\mathfrak{B}_{d}(H)$ permute
the ends of ${\mathcal{C}}_{d}(O,Y)$. As a consequence, this answers Question
5.37 of [AV20].

As we explain next, the group $\mathfrak{B}_{d}(H)$ has the same finiteness
properties as $H$. Recall that a group $G$ is of _type $F_{n}$_ if there
exists an aspherical CW-complex whose fundamental group is $G$ and whose
$n$-skeleton is finite; a group is of _type $F_{\infty}$_ if it is of type
$F_{n}$ for all $n\geq 1$. Our main result is:

######  Theorem 1.1.

Let $O$ be a compact orientable surface, and $Y$ a closed surface of genus
$\leq 1$. Then $\mathfrak{B}_{d}(H)$ is of type $F_{n}$ if and only if $H$ is.

######  Remark 1.2.

A minor variation of our construction produces a slightly larger family
$\mathfrak{B}_{d,r}$ of groups, which surject onto the more general symmetric
Thompson groups $V_{d,r}(G)$; see Remark 3.1 for details. Theorem 1.1 above is
still true in this marginally broader setting, by a word-for-word copy of our
arguments below.

Applications. We now give an application of the above theorem. Write
$\Sigma_{g}$, with $0\leq g\leq\infty$, for the closed surface of genus $g\geq
0$ with a Cantor set removed (for $g<\infty$) or the blooming Cantor tree (for
$g=\infty$); see Section 3. By the classification of infinite-type surfaces
[Ker23, Ric63], we know that ${\mathcal{C}}_{d}(O,Y)$ is homeomorphic to
$\Sigma_{g}$, for some $g$; see Remark 3.2.

When $Y$ is the sphere, $\operatorname{Map}(Y^{d})$ contains the pure braid
group with $d$ strings. By a theorem of Zaremsky [Zar17],
$\operatorname{Map}(Y^{d})$ contains a subgroup of type $F_{n}$ but not of
type $F_{n+1}$ for $d\geq n+3$. By choosing an appropriate subsurface, the
same is true when $Y$ is a torus. Selecting these subgroups as local group
$H$, we obtain:

######  Corollary 1.3.

For every $n\geq 1$ and every $g\geq\mathbb{N}\cup\\{\infty\\}$,
$\operatorname{Map}(\Sigma_{g})$ contains a subgroup $G$ of type $F_{n}$ but
not $F_{n+1}$ such that $G$ contains the mapping class group of every compact
connected surface of genus $\leq g$ with non empty boundary.

We remark that the above corollary also follows from the combination of the
main result of [Ara+21] with the diagonal labeling of our construction; see
Remark 3.8. As both families of groups contain the asymptotic mapping class
groups, they are dense inside $\operatorname{Map}(\Sigma_{g})$ when $g<\infty$
[SW21a, Theorem 3.19].

Acknowledgements. We would like to thank Louis Funar for conversations. We are
grateful to Chris Leininger for an argument that fixes an error in Proposition
6.6 in a previous version; see Remark 6.11.

##  2\. Preliminaries

In this section we provide the background material needed in our arguments. We
refer the reader to [FM12] and [AV20], respectively, for details on mapping
class groups of finite and infinite type surfaces.

###  2.1. Surfaces

Throughout, all surfaces will be connected, orientable, and second countable.
If a surface $S$ has non empty boundary, we will further assume that $\partial
S$ has finitely many connected components. We say that $S$ has finite type if
$\pi_{1}(S)$ is finitely generated; otherwise it has infinite type. An end of
$S$ is an element of

$\varprojlim\pi_{0}(S\setminus K),$

where the inverse limit is taken over all compact sets $K\subset S$, directed
with respect to inclusion. We say that an end is planar if it has a
neighborhood of genus zero; otherwise it is non-planar. Denote the set of ends
by $\operatorname{Ends}(S)$ and its subset of non-planar ends by
$\operatorname{Ends}_{np}(S)$.

The set of ends $\operatorname{Ends}(S)$ becomes a topological space by
endowing it with the limit topology arising from the discrete topology on each
of the terms defining the inverse limit above. Equipped with this topology,
$\operatorname{Ends}(S)$ is homeomorphic to a closed subspace of the Cantor
set, and $\operatorname{Ends}_{np}(S)$ is a closed subset of
$\operatorname{Ends}(S)$. By work of Kerékjártó [Ker23] and Richards [Ric63],
two surfaces are homeomorphic if and only if they have the same genus and
number of boundary components, and there is a homeomorphism between their
space of ends that restricts to a homeomorphism between the subspaces of non-
planar ends; see [Ric63] for a precise statement.

For $g\in\mathbb{N}\cup\\{0\\}$, we denote by $\Sigma_{g}$ the connected
orientable surface without boundary, of genus $g$ and with a Cantor space of
ends, all of them planar; we remark that $\Sigma_{0}$ is usually referred to
as the Cantor tree surface. Similarly, let $\Sigma_{\infty}$ be the blooming
Cantor tree surface, namely the connected orientable surface with empty
boundary, infinite genus, and whose space of ends is a Cantor set and every
end is non-planar. Finally, for $0\leq g\leq\infty$ and $b\geq 1$ write
$\Sigma_{g}^{b}$ for the surface that results by removing $b$ open discs from
$\Sigma_{g}$.

###  2.2. Curves

By a curve on $S$ we mean the isotopy class of a simple closed curve on $S$.
We say that a curve on $S$ is essential if it does not bound a disc or an
annulus whose other boundary component is a boundary component of $S$.

###  2.3. Mapping class group

Given a surface $S$, write $\operatorname{Homeo}^{+}(S,\partial S)$ for the
group of orientation preserving self-homeomorphisms of $S$ that restrict to
the identity on $\partial S$. The mapping class group of $S$ is

$\operatorname{Map}(S):=\pi_{0}\left(\operatorname{Homeo}^{+}(S,\partial
S)\right).$

We will often blur the distinction between a homeomorphism and its isotopy
class; however, if we want to make this distinction explicit, we will use
$[f]$ to denote the isotopy class of the homeomorphism $f$.

The mapping class group $\operatorname{Map}(S)$ is countable if and only if
$S$ has finite type. When $S$ has infinite type, we endow
$\operatorname{Map}(S)$ with the quotient topology of the compact-open
topology on $\operatorname{Homeo}^{+}(S,\partial S)$; with respect to this
topology, $\operatorname{Map}(S)$ is a zero-dimensional, non-locally compact
group; see [AV20].

We will also need a version of the mapping class group that allows boundary
components to be permuted. To this end, for each boundary circle
$a\subset\partial S$ we fix a homeomorphism $p_{a}:a\to\mathbb{S}^{1}$, which
we call a parametrization of $a$. We say that an orientation preserving
homeomorphism $f:S\to S$ respects the boundary parametrization if
$p_{f(a)}\circ f_{|a}\equiv p_{a}$ for all $a\in\pi_{0}(\partial S)$. We now
set:

######  Definition 2.1 (Boundary-permuting mapping class group).

The boundary permuting mapping class group $\operatorname{Map}_{\partial}(S)$
is the group of isotopy classes of orientation-preserving self-homeomorphisms
of $S$ that respect the boundary parametrization, modulo isotopies that fix
$\partial S$ pointwise.

######  Remark 2.2.

Since we are assuming that surfaces have a finite number of boundary
components, $\operatorname{Map}(S)$ is a (normal) finite-index subgroup of
$\operatorname{Map}_{\partial}(S)$; moreover, the two groups coincide if and
only if $S$ has at most one boundary component.

Finally, given a surface $S$ and $A\subseteq\pi_{0}(\partial S)$, we define
$\operatorname{Map}_{\partial}(S,A)$ to be the subgroup of
$\operatorname{Map}_{\partial}(S)$ whose elements restrict to the identity on
every element of $A$. Of course, if $A=\partial S$, then
$\operatorname{Map}_{\partial}(S,A)=\operatorname{Map}(S)$.

##  3\. Cantor surfaces and block mapping class groups

We now recall the definition of a Cantor surface from [Ara+21]. Let $O$ be an
arbitrary compact orientable surface, and $Y$ homeomorphic to a sphere or a
torus. Denote by $Y^{d}$ the surface obtained from $Y$ by removing $d+1$ open
disks. Write $b_{0},\ldots,b_{d}$ for the boundary components of $Y^{d}$; we
will refer to $b_{0}$ as the top boundary component of $Y^{d}$. For each
$i=1,\ldots,d$, we choose an orientation-reversing homeomorphism
$\mu_{i}:b_{0}\to b_{i}$. We inductively define a family $\\{O_{n}\\}$ of
compact surfaces as follows:

  1. (1)

$O_{1}$ is the surface that results by removing an open disk from $O$.

  2. (2)

$O_{2}$ is the surface obtained by gluing (with respect to some pre-fixed
orientation-reversing homeomorphism) a copy of $Y^{d}$, along its top boundary
component, to the boundary of $O_{1}$ created in the previous step;

  3. (3)

For each $n\geq 2$, $O_{n}$ is the surface obtained by gluing a copy of
$Y^{d}$, along its top boundary component, to each of the boundary components
of $O_{n-1}\setminus O_{n-2}$, using the homeomorphisms $\mu_{i}$ above.

The Cantor surface ${\mathcal{C}}_{d}(O,Y)$ is the union of the surfaces
$O_{k}$ above. The closure of each of the connected components of
$O_{n}\setminus O_{n-1}$ is called a block of ${\mathcal{C}}_{d}(O,Y)$; by
construction, every block is homeomorphic to $Y^{d}$. The top boundary
$\partial_{T}B$ of a block $B$ is the image of the top boundary of $Y^{d}$
under the gluing map.

$O$
---
$Y^{2}$
---
$C_{2}(O,Y)$
---
Figure 1. Example of a Cantor surface

######  Remark 3.1.

Following [SW21a] and [Ara+21], one may slightly generalize the above
construction by first removing $r\geq 1$ open discs from $O$ and then gluing
copies of $Y^{d}$ to each of these boundary components. Our main result has a
direct translation to the case of the surfaces ${\mathcal{C}}_{d,r}(O,Y)$
obtained in this way; however, we have decided to restrict our attention to
the case $r=1$ in order to minimize the notation and level of technicality.

A suited subsurface of ${\mathcal{C}}_{d}(O,Y)$ is a connected subsurface that
is the union of $O_{1}$ and finitely many blocks; in particular, a suited
subsurface is always compact. A suited curve of ${\mathcal{C}}_{d}(O,Y)$ is a
boundary component of a suited subsurface that is not one of the original
boundary components of $O$. Given a suited subsurface $Z$, we refer to the
boundary curves of $Z$ that are suited curves of ${\mathcal{C}}_{d}(O,Y)$ as
the suited boundary components of $Z$; the rest (if any) are called the
primary boundary components of $Z$. Given a suited boundary component
$a\subset\partial Z$, we say that a connected component
$X\subset{\mathcal{C}}_{d}(O,Y)\setminus Z$ is adjacent to $a$ if the
intersection of the closures of $Z$ and $X$ is equal to $a$.

######  Remark 3.2.

The space of ends of ${\mathcal{C}}_{d}(O,Y)$ is always homeomorphic to a
Cantor set, and every two ends of ${\mathcal{C}}_{d}(O,Y)$ are related by a
homeomorphism of ${\mathcal{C}}_{d}(O,Y)$. In particular, by the
classification of non-compact surfaces of Kerékjártó-Richards [Ric63], one
has:

  * •

If $Y\cong{\mathbb{S}}^{2}$, then ${\mathcal{C}}_{d}(O,Y)\cong\Sigma_{g}^{b}$,
where $g$ and $b$ are, respectively, the genus and number of boundary
components of $O$.

  * •

If $Y\cong{\mathbb{S}}^{1}\times{\mathbb{S}}^{1}$, then
${\mathcal{C}}_{d}(O,Y)\cong\Sigma_{\infty}^{b}$, where $b$ is the number of
boundary components of $O$.

For the sake of notational convenience, for the rest of the section we will
denote ${\mathcal{C}}_{d}(O,Y)$ simply by ${\mathcal{C}}_{d}$, unless we need
to make explicit reference to the surfaces $O$ and $Y$.

######  Definition 3.3 (Block homeomorphism).

Let ${\mathcal{C}}_{d}$ be a Cantor surface. We say that a homeomorphism
$f:{\mathcal{C}}_{d}\to{\mathcal{C}}_{d}$ is block if there exists a suited
subsurface $Z\subset{\mathcal{C}}_{d}$ such that:

  1. (1)

$f(Z)$ is also suited;

  2. (2)

The restriction $f:{\mathcal{C}}_{d}\setminus Z\to{\mathcal{C}}_{d}\setminus
f(Z)$ sends blocks to blocks.

The suited subsurface $Z$ in the definition above is called a defining surface
for $f$. Note that any surface containing a defining surface for $f$ is itself
a defining surface for $f$. As an immediate consequence, the composition of
two block homeomorphisms is also block; in particular, the set of block
homeomorphisms of ${\mathcal{C}}_{d}$ is in fact a group under composition.

Next, for each block $B\subset{\mathcal{C}}_{d}$ we fix, once and for all, a
homeomorphism $\iota_{B}:B\to Y^{d}$ that takes the top boundary component of
$B$, denoted $\partial_{T}B$, to $b_{0}$, namely the top boundary component of
$Y^{d}$. The homeomorphism $\iota_{B}$ induces an isomorphism (abusing
notation)

$\iota_{B}:\operatorname{Map}_{\partial}(B,\partial_{T}B)\to\operatorname{Map}_{\partial}(Y^{d},b_{0}),$

and we will use $\iota_{B}$ to identify
$\operatorname{Map}_{\partial}(B,\partial_{T}B)$ with
$\operatorname{Map}_{\partial}(Y^{d},b_{0})$ without further mention. Note
that $\iota_{B}$ induces an order on the remaining boundaries of $B$ from that
of $Y^{d}$.

######  Definition 3.4 ($H$-block homeomorphism).

Let $H<\operatorname{Map}_{\partial}(Y^{d},b_{0})$, and
$f:{\mathcal{C}}_{d}\to{\mathcal{C}}_{d}$ be a block homeomorphism with
defining surface $Z$. We say that $f$ is $H$-block if for every suited
boundary component $a\subset\partial Z$ there is an element $h_{a}\in H$ such
that, for every block $B$ in the connected component of
${\mathcal{C}}_{d}\setminus Z$ adjacent to $a$, we have that

$[\iota_{f(B)}\circ f\circ\iota_{B}^{-1}]\equiv h_{a}.$

We will say that the boundary curve $a$ is labeled by the element $h_{a}$,
that $f$ is _labeled by the tuple_ $\mathbf{h}=(h_{1},\ldots,h_{n})$ outside
of $Z$ (where $n$ is the number of suited boundary components of $Z$), and
that $f$ eventually acts like $\mathbf{h}$.

######  Definition 3.5 ($H$-block mapping class group).

Let ${\mathcal{C}}_{d}$ be a Cantor surface, and
$H<\operatorname{Map}_{\partial}(Y^{d},b_{0})$. The block mapping class group
with local group $H$, denoted $\mathfrak{B}_{d}(H)$, is the group of isotopy
classes of $H$-block homeomorphisms of ${\mathcal{C}}_{d}$.

In the case when $H=\\{1\\}$, we will simply write
$\mathfrak{B}_{d}=\mathfrak{B}_{d}(H)$. We remark that this case corresponds
to the asymptotic mapping class groups of [Ara+21] which, as mentioned in the
introduction, are in turn an abstraction of the ones considered by Funar,
Kapoudjian and the first named author [FK04, FK09, AF21].

######  Remark 3.6.

The group $\mathfrak{B}_{d}(H)$ is always countable, as
$\operatorname{Map}(Y,b_{0})$ is countable and so is the set of suited
subsurfaces of ${\mathcal{C}}_{d}(O,Y)$. In stark contrast, the group of
isotopy classes of block homeomorphisms of ${\mathcal{C}}_{d}(O,Y)$ is
uncountable.

######  Remark 3.7.

Considering the surfaces ${\mathcal{C}}_{d,r}(O,Y)$ as in Remark 3.1 would
give rise to more general block mapping class groups
$\mathfrak{B}_{d,r}(O,Y)$. As will become apparent, the obvious adaptation of
our arguments below would yield the analog of Theorem 1.1 for this (a priori,
larger) set of groups.

###  3.1. An alternate description

We now give a useful alternate representation of elements of
$\mathfrak{B}_{d}(H)$, reminiscent of the tree-pair representation of elements
of (symmetric) Higman-Thompson groups; see Section 4.

Let $f\in\mathfrak{B}_{d}(H)$, and consider a defining subsurface
$Z\subset{\mathcal{C}}_{d}$ for $f$, with suited boundary components $a_{i}$,
$i=1,\ldots,n$. Write $X_{i}$ for the connected component of
${\mathcal{C}}_{d}\setminus Z$ adjacent to $a_{i}$, and denote by $f_{Z}$
(resp. $h_{i}$) the restriction of $f$ to $Z$ (resp. the element of $H$ that
$f$ acts like on $X_{i}$). With this notation, $f$ is determined by the tuple

$\left(Z,(f_{Z},\mathbf{h}),f(Z)\right),$

where $\mathbf{h}=(h_{1},\ldots,h_{n})$. Finally, observe that, since defining
surfaces are not unique, there are countably many different such tuples
representing the same element of $\mathfrak{B}_{d}(H)$. We use
$[Z,(f_{Z},\mathbf{h}),f(Z)]$ to denote the equivalence class (with respect to
the obvious equivalence relation) containing
$\left(Z,(f_{Z},\mathbf{h}),f(Z)\right)$.

######  Remark 3.8.

If one takes the labeling to be the same for all components, i.e.
$\mathbf{h}=(h,\cdots,h)$ for some $h\in H$, then the set of all such elements
forms a subgroup $\mathfrak{DB}_{d}(H)$. Since the surjection
$\mathfrak{DB}_{d}(H)\to H$ mapping $[Z,(f_{Z},\mathbf{h}),f(Z)]$ to $h$
splits, we have an isomorphism

$\mathfrak{DB}_{d}(H)\cong\mathfrak{B}_{d}(\\{1\\})\rtimes H.$

Now $\mathfrak{B}_{d}(\\{1\\})$ is of type $F_{\infty}$ [Ara+21], hence
$\mathfrak{B}_{d}(\\{1\\})\rtimes H$ if of type $F_{n}$ if and only if $H$ is
[Geo08, Section 7.2].

##  4\. The relation with symmetric Higman-Thompson groups

In this section we will relate our groups to the so-called symmetric Higman-
Thompson groups, introduced by Farley and Hughes in [FH15]. We commence by
reviewing the construction of symmetric Higman-Thompson groups; see [FH15] for
a detailed treatment.

Let $\mathcal{T}_{d}$ be the infinite rooted tree of degree $d\geq 2$, where
every vertex other than the root has degree $d+1$. We endow $\mathcal{T}_{d}$
with the path-metric that arises by deeming each edge to have length one; with
respect to this path-metric, the end space of $\mathcal{T}_{d}$ is naturally
homeomorphic with the set of geodesics emanating from the root, which in turn
is homeomorphic to the standard Cantor set $\mathfrak{C}$.

Given a vertex $v\in\mathcal{T}_{d}$ at distance $k\geq 0$ from the root, we
say that the vertex $w$ is a descendant of $v$ if $d(v,w)=1$ and $w$ is at
distance $k+1$ from the root. A subtree $F$ of $\mathcal{T}_{d}$ is complete
if, for every vertex $v\in F$, whenever $F$ contains a descendant of $v$, it
contains all of its descendants. Observe that if $F$ is a complete finite
rooted subtree of $\mathcal{T}_{d}$ then, for every $c\in\mathfrak{C}$ there
is a unique leaf $l$ of $F$ such that the geodesic in $\mathcal{T}_{d}$
representing $c$ contains $l$; we say that $l$ is the leaf of $F$ nearest to
$c$.

###  4.1. Higman-Thompson groups

Let $F_{-}$ and $F_{+}$ be complete finite rooted subtrees of
$\mathcal{T}_{d}$, with the same number of leaves, and $b$ a bijection between
their sets of leaves. Then the triple $(F_{-},b,F_{+})$ extends to a
homeomorphism $g:\mathfrak{C}\to\mathfrak{C}$ by identifying (in the obvious
way) the infinite subtree of $\mathcal{T}_{d}$ below the leaf $l$ with that
below the leaf $b(l)$. We say that the triple $(F_{-},b,F+)$ is a tree-pair
representative of $g$; of course, there are infinitely many tree-pair
representatives of any given homeomorphism.

A pleasant exercise shows that if two self-homeomorphisms of $\mathfrak{C}$
admit a tree-pair representative, then the same is true about their
composition. Thus, we define the Higman-Thompson group $V_{d}$ [Hi74] as the
group of homeomorphisms of $\mathfrak{C}$ that have a tree-pair
representative; we remark that $V_{2}$ is the classical Thompson group $V$. A
well-known fact [Bro87] is that $V_{d}$ is of type $F_{\infty}$.

###  4.2. Symmetric Thompson groups

Next, we recall the definition of the symmetric versions of the Higman-
Thompson groups, introduced by Farley and Hughes [FH15].

Enumerate the descendants of every vertex with $\\{0,\dots,d-1\\}$. Since the
(Cantor) space of ends of $\mathcal{T}_{d}$ can be identify with the set of
geodesic of $\mathcal{T}_{d}$, we can also identify it with the set of
infinite sequences $(t_{1},t_{2},t_{3},\ldots)$, where $0\leq t_{i}<d$ for
every $i\geq 1$.

Let $F_{-}$ and $F_{+}$ be complete finite subtrees of $\mathcal{T}_{d}$, with
the same number of leaves, and $b$ a bijection between their sets of leaves.
Write $l_{i}$, $i=1,\ldots,n$ for the leaves of $F_{-}$.

Let $G$ be a subgroup of the symmetric group $\operatorname{Sym}(d)$ on $d$
elements. With respect to the above notation, for every leaf $l_{i}$ we chose
$g_{i}\in G$, and write $\mathbf{g}=\\{g_{1},\ldots,g_{n}\\}$. Then the tuple

$(F_{-},(b,\mathbf{g}),F_{+})$

induces a homeomorphism $\psi:\mathfrak{C}\to\mathfrak{C}$ as follows. Let
$c\in\mathfrak{C}$ be given in terms of an infinite sequence
$(c_{1},c_{2},\dots)$, and consider the leaf $l_{i}$ of $F_{-}$ nearest to
$c$, so that $(c_{1},c_{2},\dots,c_{m})$ represents the leaf $l_{i}$. Then the
leaf $b(l_{i})$ is represented by a finite sequence $b(l_{i})$ (using a slight
abuse of notation), and define

$\psi(c)=(b(l_{i}),(g_{i}(c_{m+1}),g_{i}(c_{m+2}),\dots).$

We say that $\psi$ eventually acts like $\mathbf{g}$, and the tuple
$(F_{-},(b,\mathbf{g}),F_{+})$ is called the symmetric tree-pair
representative of $\psi$; as before, there are infinitely many symmetric tree-
pair representatives of the same homeomorphism of $\mathfrak{C}$.

Again, if two self-homeomorphisms of $\mathfrak{C}$ admit a tree-pair
representative, then the same is true about their composition. Hence we define
the symmetric Higman-Thompson group $V_{d}(G)$ with local group $G$ as the
group of homeomorphisms of $\mathfrak{C}$ that admit a $G$ tree-pair
representative. Observe that if $G=\\{1\\}$ then $V_{d}(G)=V_{d}$.

Finally, we note that the groups $V_{d}(G)$ are examples of FSS groups in the
sense of Farley and Hughes, which are all of type $F_{\infty}$ [FH15].

######  Remark 4.1.

The groups $V_{d}(G)$ are part of a larger family of symmetric Higman-Thompson
groups $V_{d,r}(G)$ groups, where one allows $r$-forests of $d$-valent rooted
trees instead of just $d$-valent rooted trees. These groups are also FFS
groups in the sense of Farley and Hughes [FH15]; as such, they are of type
$F_{\infty}$.

###  4.3. A surjective homomorphism

Recall from Section 3 that the Cantor surface ${\mathcal{C}}_{d}(O,Y)$ is the
union of compact surfaces $O_{k}$, with $k\geq 1$.

We construct an abstract simplicial graph $\Gamma$ as follows: the vertices of
$\Gamma$ are the blocks of ${\mathcal{C}}_{d}(O,Y)$ plus $O_{1}$, and two
vertices are adjacent in $\Gamma$ if and only if the corresponding (suited)
subsurfaces of ${\mathcal{C}}_{d}(O,Y)$ share one suited boundary component.
Observe that, by construction, $\Gamma$ is an infinite rooted $d$-ary tree,
with the root corresponding to $O_{1}$; in particular, $\Gamma$ is isomorphic
to $\mathcal{T}_{d}$. Fixing one such isomorphism, there is an obvious map

$\tau:{\mathcal{C}}_{d}(O,Y)\to\mathcal{T}_{d}$

such that the preimage of the root is $O_{1}$, and the preimage of the
midpoint of every edge of $\mathcal{T}_{d}$ is a suited curve. Next, there is
a natural surjective homomorphism

$q:\operatorname{Map}(Y^{d},b_{0})\to\operatorname{Sym}(d)$

which records the manner in which the elements of
$\operatorname{Map}(Y^{d},b_{0})$ permute the boundary components of $Y^{d}$.

Let $H<\operatorname{Map}(Y^{d},b_{0})$ and write $\bar{H}$ for the $q$-image
of $H$. Let $f\in\mathfrak{B}_{d}(H)$, and choose a tree-pair representative
$(Z,(f_{Z},\mathbf{h}),f(Z))$ for $f$. Then $f$ induces a bijection $b_{f}$
between the leaves of $\tau(Z)$ and $\tau(f(Z))$; also, on the connected
component $X_{i}$ of ${\mathcal{C}}_{d}(O,Y)\setminus Z$, the homeomorphism
$f$ induces, via $\tau$, an isomorphism
$f^{*}_{X_{i}}:\tau(X_{i})\to\tau(f(X_{i}))$ that acts like $q(h_{i})$. As
such, the tuple

$\left(\tau(X),(b_{f},q(\textbf{h})),\tau(f(X)\right)$

is an element of $V_{d}(\bar{H})$. An exercise in the definitions of tree-pair
representatives shows that the map

$(Z,(f_{Z},\mathbf{h}),f(Z))\mapsto\left(\tau(X),(b_{f},q(\textbf{h})),\tau(f(X)\right),$

descends to a map

$\pi:\mathfrak{B}_{d}(H)\to V_{d}(\bar{H}),$

which is in addition a homomorphism. Moreover, one has:

######  Proposition 4.2.

The homomorphism $\pi:\mathfrak{B}_{d}(H)\to V_{d}(\bar{H})$ is surjective.

###### Proof.

Consider a tree-pair representative $(F_{-},(b,\mathbf{g}),F_{+})$ of an
element of the group $V_{d}(\bar{H})$, and denote by $F_{-}^{\prime}$ (resp
$F^{\prime}_{+}$) the result of removing from $F_{-}$ (resp. $F_{+}$) their
leaves and the open half edges incident on these leaves. We know that
$Z_{-}:=\tau^{-1}(F^{\prime}_{-})$ and $Z:=\tau^{-1}(F^{\prime}_{+})$ are
suited subsurfaces of $\mathcal{C}_{d}(O,Y)$. Let $\hat{f}:Z_{-}\to Z_{+}$ be
a homeomorphism such that the induced bijection between the sets of boundary
components of $Z_{-}$ and $Z_{+}$ coincides with $b$. Writing
$X_{1},\ldots,X_{n}$ for the connected components of ${\mathcal{C}}_{d}(O,Y)$,
for each $i$ we choose an element $h_{i}\in H$ with $q(h_{i})=g_{i}$. Then,
the element $f\in\mathfrak{B}_{d}(H)$ with tree-pair representative

$(Z_{-},(\hat{f},\mathbf{h}),Z_{+})$

maps to $(F_{-},(b,\mathbf{g}),F_{+})\in V_{d}(\bar{H})$ under $\pi$, as
desired. $\square$

######  Remark 4.3.

After Remarks 3.7 and 4.1, a direct adaptation of the arguments above yield
the existence of a surjective homomorphism

$\mathfrak{B}_{d,r}(H)\to V_{d,r}(\bar{H}),$

where $V_{d,r}(\bar{H})$ is the symmetric Higman-Thompson group with $r$
roots.

######  Remark 4.4.

Although it is not very relevant to our discussion, it is worth mentioning
that the group of block homeomorphisms of $\mathcal{C}_{d}(O,Y)$ surjects onto
the Neretin group of [Ner92].

##  5\. Finiteness properties

###  5.1. The complex

In this section, we will construct an infinite-dimensional contractible cube
complex ${\mathfrak{X}}_{d}(O,Y,H)$ on which $\mathfrak{B}_{d}(H)$ acts
cellularly, closely related to the classical Stein–Farley complexes for
Higman–Thompson groups [Ste92, Far03], and very much inspired by the complex
constructed in [GLU20, SW21, Ara+21].

Here the surfaces $O$ and $Y$ and the integer $d$ will play virtually no role
in the discussion, and so we will simply write
${\mathcal{C}}={\mathcal{C}}_{d}(O,Y)$ and
${\mathfrak{B}}={\mathfrak{B}}_{d}(H)$. We remark that the definitions and
results in this section are direct translations of those in [Ara+21, Section
5]; we include short proofs for completeness.

###  5.2. The Stein complex

Consider all ordered pairs $(Z,f)$ where $Z$ is a suited subsurface of
${\mathcal{C}}$ and $f\in{\mathfrak{B}}$. We will say a pair $(Z_{1},f_{1})$
is equivalent to a pair $(Z_{2},f_{2})$, written
$(Z_{1},f_{1})\sim(Z_{2},f_{2})$, if and only if there are representing
homeomorphisms $f_{1}$ and $f_{2}$ (abusing notation) such that
$f_{2}^{-1}\circ f_{1}$ maps $Z_{1}$ to $Z_{2}$ where $Z_{1}$ is a defining
surface for $f_{2}^{-1}\circ f_{1}$ as an $H$-block homeomorphism. A
straightforward calculation confirms that this relation is symmetric,
reflexive, and transitive, i.e. an equivalence relation.

We will denote by $[Z,f]$ the equivalence class containing $(Z,f)$ with
respect to this relation, and write ${\mathcal{S}}$ for the set of all
equivalence classes. Importantly, the group ${\mathfrak{B}}$ acts on
${\mathcal{S}}$ via left multiplication as follows:

$g\cdot[Z,f]=[Z,g\circ f].$

Recall that by definition, a suited subsurface is the union of $O_{1}$ and
finitely many additional blocks. We define the _complexity_ $h(Z,f)$ of
$(Z,f)$ to be the number of blocks in $Z$. We observe that if
$(Z_{1},f_{1})\sim(Z_{2},f_{2})$ then $h(Z_{1},f_{1})=h(Z_{2},f_{2})$ and thus
$h$ gives a well-defined function $h:{\mathcal{S}}\rightarrow{\mathbb{N}}$.

######  Definition 5.1.

The _complexity_ of $[Z,f]\in{\mathcal{S}}$ is $h(Z,f)$ for some, and hence
any, representative of $[Z,f]$.

Let $x_{1}$ and $x_{2}$ be two classes in ${\mathcal{S}}$ and define
$x_{1}\leq x_{2}$ if and only if $x_{1}=[Z_{1},f]$ and $x_{2}=[Z_{2},f]$ for
suited subsurfaces with $Z_{1}\subseteq Z_{2}$. When the inclusion is proper,
we write $x_{1}<x_{2}$. The following is a word-for-word adaptation of Lemmas
5.2 and 5.3 of [Ara+21]:

######  Lemma 5.2.

$({\mathcal{S}},\leq)$ is a directed poset. Moreover, if $x<y\in{\mathcal{S}}$
then $h(x)<h(y)$. In particular, its geometric realization $|{\mathcal{S}}|$
is contractible

Following the same strategy of [Ara+21], we now consider a second, finer
relation $\preceq$ on ${\mathcal{S}}$. Let $x_{1},x_{2}$ be two vertices in
${\mathcal{S}}$. We will say $x_{1}\preceq x_{2}$ if $x_{1}=[Z,f]$ and
$x_{2}=[Z\cup X,f]$ where $X$ is a disjoint union of blocks. As usual, if
$x_{1}\preceq x_{2}$ and $x_{1}\neq x_{2}$, we will write $x_{1}\prec x_{2}$.
This finer order is no longer a partial ordering as it is not transitive.
Nevertheless, it is true that if $x_{1}\preceq x_{3}$ and $x_{1}\leq x_{2}\leq
x_{3}$ then $x_{1}\preceq x_{2}\preceq x_{3}$. Thus we can define a simplex
$x_{0}<x_{1}<\cdots<x_{k}$ in $|{\mathcal{S}}|$ to be _elementary_ if
$x_{0}\preceq x_{k}$.

######  Definition 5.3 (Stein complex).

The _Stein complex_ ${\mathfrak{X}}={\mathfrak{X}}(O,Y,H)$ is the full
subcomplex of $|{\mathcal{S}}|$ consisting of elementary simplices.

We note that the Stein complex ${\mathfrak{X}}$ is more tractable than
$|{\mathcal{S}}|$ but still has many of the properties we care about. For
instance, the action of ${\mathfrak{B}}$ on ${\mathcal{S}}$ preserves the
relation $\preceq$, so thus ${\mathfrak{B}}$ has a simplicial action on
${\mathfrak{X}}$. Moreover, a classical result of Quillen [Qui78] implies the
following, whose proof is a verbatim copy of Lemmas 5.6 and 5.7 of [Ara+21]:

######  Proposition 5.4.

The Stein complex ${\mathfrak{X}}$ is contractible.

###  5.3. The Stein-Farley cube complex

In this section, we group the simplices of ${\mathfrak{X}}$ together to form a
cube complex. Let $x$ and $y$ be vertices of ${\mathfrak{X}}$ with $x\leq y$,
and set $[x,y]=\\{z\mid x\leq z\leq y\\}$. The same argument of [Ara+21, Lemma
5.8] shows that $[x,y]$ has the structure of a finite Boolean lattice; in
particular, its geometric realization is (the barycentric subdivision of) a
cube of dimension $h(y)-h(x)$. Moreover, every two cubes intersect over a
face, by the same argument as in [Ara+21, Lemma 5.10]. In particular, we have
(see [Ara+21, Lemma 5.11]):

######  Proposition 5.5.

The complex ${\mathfrak{X}}$ has the structure of a cube complex with each
cube defined by an interval $[x,y]$ with $x\preceq y$.

######  Definition 5.6 (Stein-Farley complex).

The complex ${\mathfrak{X}}$ equipped with the above cubical structure is
called the _Stein-Farley cube complex_ associated to the Cantor surface
${\mathcal{C}}$ and the subgroup
$H<\operatorname{Map}_{\partial}(Y^{d},b_{0})$.

##  6\. Brown’s Criterion and Discrete Morse Theory on the Stein-Farley cube
complex

As was the case in [Ara+21], now that we have an action of ${\mathfrak{B}}$ on
the Stein-Farley complex, we will want to apply Brown’s criterion to deduce
the finiteness properties [Bro87]. First, recall that a filtration
$\\{K_{j}\\}_{j\geq 1}$ of a space $K$ is called _essentially $n$-connected_
if for every $i\geq 1$, there exists an $i^{\prime}\geq i$ such that
$\pi_{\ell}(K_{i}\rightarrow K_{i^{\prime}})$ is trivial for all $\ell\leq n$.

We remark that the definitions and results in this Section are direct
translations of those in [Ara+21, Section 6]; we include short proofs for
completeness.

######  Theorem 6.1 (Brown’s Criterion).

Let $n$ be in ${\mathbb{N}}$ and assume that a group $G$ acts on an
$(n-1)$-connected CW-complex ${\mathcal{K}}$. Assume that the stabilizer of
every $k$-cell of ${\mathcal{K}}$ is of type $F_{n-k}$. Let
$\\{{\mathcal{K}}_{j}\\}_{j\geq 1}$ be a filtration of ${\mathcal{K}}$ such
that each ${\mathcal{K}}_{j}$ is finite modulo the action of $G$. Then $G$ is
of type $F_{n}$ if and only if $\\{{\mathcal{K}}_{j}\\}_{j}$ is essentially
$(n-1)$-connected.

First, observe that the complexity function given in Definition 5.1 gives a
function $h:{\mathfrak{X}}^{(0)}\rightarrow{\mathbb{N}}$ which naturally
extends affinely to each cube to obtain a _height function_

$h:{\mathfrak{X}}\rightarrow{\mathbb{R}},$

which is ${\mathfrak{B}}$-equivariant with respect to the trivial action of
${\mathfrak{B}}$ on ${\mathbb{R}}.$ Each cube has a unique vertex at which $h$
is maximized (respectively minimized) which we call the the _top_
(respectively _bottom_) of the cube. Note that this agrees with our definition
of top and bottom above. Now, for any integer $k$, we define
${\mathfrak{X}}^{\leq k}$ to be the subcomplex of ${\mathfrak{X}}$ spanned by
vertices with height at most $k$. An obvious adaptation of [Ara+21, Lemma 6.2]
yields:

######  Lemma 6.2.

The group ${\mathfrak{B}}$ acts cocompactly on ${\mathfrak{X}}^{\leq k}$ for
all $1\leq k<\infty$.

Next, we prove:

######  Lemma 6.3.

Let $C$ be a cube in ${\mathfrak{X}}$ with bottom vertex given by $x=[Z,g]$.
Then the stabilizer of $C$ in ${\mathfrak{B}}_{d}(H)$ is isomorphic to a
finite index subgroup of $\operatorname{Map}_{\partial}(Z)\ltimes H^{m}$ where
$m$ is the number of suited boundary components of $Z$.

###### Proof.

Fix a cube $C$ in ${\mathfrak{X}}$ and let $x=[Z,g]$ be its bottom vertex. We
may assume without loss of generality that $g={{\rm id}}$. The cube $C$ is
spanned by vertices obtained by attaching blocks to a collection $A$ of suited
boundary components. Now any element $f\in{\mathfrak{B}}$ stabilizes the cube
if and only if it both fixes $x$ and leaves $A$ invariant. The element $f$
fixes $x=[Z,{{\rm id}}]$ if and only if $f$ sends $Z$ to $Z$ and is labeled by
a tuple $\bf h$ outside of $Z$ and so it is therefore a subgroup of
$\operatorname{Map}_{\partial}(Z)\ltimes H^{m}$. Moreover
$\operatorname{Map}(Z)\ltimes H^{m}$ preserves the cube and so the result
follows from Remark 2.2.

$\square$

Since mapping class groups of compact surfaces are of type $F_{\infty}$
[Har71], and being of type $F_{n}$ is invariant under short exact sequences
and passing to finite-index subgroups [Geo08, Section 7.2], the previous lemma
implies:

######  Corollary 6.4.

Let $C$ be a cube in ${\mathfrak{X}}$. Then the stabilizer of $C$ in
${\mathfrak{B}}_{d}(H)$ has type $F_{n}$ if $H$ does.

###  6.1. Descending links

We have now checked all of the conditions of Brown’s Criterion except one,
namely that the filtration $\\{{\mathfrak{X}}^{\leq m}\\}_{m}$ is essentially
$(n-1)$-connected. In fact, we will show the connectivity of the pairs
$({\mathfrak{X}}^{\leq m},{\mathfrak{X}}^{\leq{m+1}})$ tends to $\infty$ as
$m$ tends to $\infty$; here, recall that a pair of spaces $(L,K)$ with
$K\subseteq L$ is $k$-connected if the inclusion map $K\hookrightarrow L$
induces an isomorphism in their respective homotopy groups $\pi_{j}$ for $j<k$
and an epimorphism in $\pi_{k}$. To establish this, we will apply discrete
Morse theory to reduce the problem to analyzing the connectivity of the
descending links.

Let ${\mathcal{K}}$ be a piecewise Euclidean cell complex, and let $h$ be a
map from the set of vertices of ${\mathcal{K}}$ to the integers, such that
each cell has a unique vertex maximizing $h$. Call $h$ a _height function_ ,
and $h(y)$ the _height_ of $y$ for vertices $y$ in ${\mathcal{K}}$. For
$t\in{\mathbb{Z}}$, define ${\mathcal{K}}^{\leq t}$ to be the full subcomplex
of ${\mathcal{K}}$ spanned by vertices $y$ with $h(y)\leq t$. Similarly,
define ${\mathcal{K}}^{<t}$ and ${\mathcal{K}}^{=t}$. The _descending star_
$\operatorname{{\operatorname{st}}^{\downarrow}}(y)$ of a vertex $y$ is
defined to be the open star of $y$ in ${\mathcal{K}}^{\leq h(y)}$. The
_descending link_ $\operatorname{{\operatorname{lk}}^{\downarrow}}(y)$ of $y$
is given by the set of “local directions” starting at $y$ and pointing into
$\operatorname{{\operatorname{st}}^{\downarrow}}(y)$, that is, the link of $y$
in $\operatorname{{\operatorname{st}}^{\downarrow}}(y)$. For more details see
[BB97]. Now the following Morse Lemma is a consequence of [BB97, Corollary
2.6].

######  Lemma 6.5 (Morse Lemma).

Let ${\mathcal{K}}$ be a piecewise Euclidean cell complex and let $h$ be a
height function on ${\mathcal{K}}$.

  1. (1)

Suppose that for any vertex $y$ with $h(y)=t$,
$\operatorname{{\operatorname{lk}}^{\downarrow}}(y)$ is $(k-1)$-connected.
Then the pair $({\mathcal{K}}^{\leq t},{\mathcal{K}}^{<t})$ is $k$-connected.

  2. (2)

Suppose that for any vertex $y$ with $h(y)\geq t$,
$\operatorname{{\operatorname{lk}}^{\downarrow}}(y)$ is $(k-1)$-connected.
Then $({\mathcal{K}},{\mathcal{K}}^{<t})$ is $k$-connected.

Recall that the height function for the Stein-Farley complex ${\mathfrak{X}}$
is the affine extension of the complexity function $h$ given in Definition
5.1.

###  6.2. From descending links to the complex of labeled blocks

In this subsection we will determine the connectivity properties of descending
links. Behind all of our arguments is Hatcher-Wahl’s notion of a complete join
complex, which we now recall:

######  Definition 6.6 (Complete join).

A simplicial map $\pi:K\to L$ between simplicial complexes is said to be a
complete join if

  * •

$\pi$ is surjective;

  * •

The restriction of $\pi$ to every simplex of $K$ is injective;

  * •

For each simplex $\sigma=\langle x_{0},\ldots,x_{p}\rangle$ of $L$, the
subcomplex of $K$ consisting of those simplices that map to $\sigma$ is equal
to the join $\pi^{-1}(x_{0})\ast\cdots\ast\pi^{-1}(x_{p})$.

By a slight abuse of notation, we will say that $K$ is a complete join over
$L$.

The interest of the above definition in our situation is the following result,
which is Proposition 3.5 of [HW10]; here, recall that a complex is weakly
Cohen-Macaulay of dimension $m$ if it is $(m-1)$-connected and the link of
every $p$-simplex is $(m-p-2)$-connected.

######  Proposition 6.7.

If $K$ is a complete join complex over a weakly Cohen-Macaulay complex $L$ of
dimension $m$, then $K$ is also weakly Cohen-Macaulay of dimension $m$.

Let $x=[X,f]$ be a vertex of ${\mathfrak{X}}$; we want to analyze the
connectivity properties of its descending link. First, we may assume that
$f={\rm id}$. Then a simplex in the descending link of $x$ is determined by
$z=[Z,g]$, where there exists a collection $\\{Y_{0},\ldots,Y_{p}\\}$ of
pairwise distinct and pairwise disjoint blocks such that
$[Z^{\prime},g]\sim[X,\rm id]$ with $Z^{\prime}=Z\cup\bigcup_{i=0}^{p}Y_{p}$.

Our goal now is to map $\operatorname{lk}^{\downarrow}(x)$ onto a weakly
Cohen-Macaulay complex in such a way that $\operatorname{lk}^{\downarrow}(x)$
is a complete join complex over it. To this end, we have the following
definition:

######  Definition 6.8 (Block Complex).

Given a suited subsurface $W$ and a (not necessarily proper) subset $A$ of
boundary components of $W$, the (unlabeled) block complex $\mathcal{P}(W,A)$
is the simplicial complex whose $p$-simplices are sets
$\\{Z_{0},\ldots,Z_{p}\\}$ of pairwise disjoint isotopy classes of subsurfaces
of $W$, each homeomorphic to $Y^{d}$, with the property that exactly one
boundary component is essential in $W$, while the rest belong to $A$.

We remark that $\mathcal{P}(W,A)$ is called the piece complex in [Ara+21]. The
following is [Ara+21, Theorem 10.1];

######  Theorem 6.9.

Let $W_{g}^{b}$ be the compact, connected surface of genus $g$ with $b$
boundary components, and let $A$ be a (not necessarily proper) subset of these
boundary components. Then $\mathcal{P}(W_{g}^{b},A)$ is weakly Cohen-Macaulay
of dimension $m$ provided that $m\leq\frac{g-1}{2}$ and
$m\leq\frac{|A|+2d}{2d-1}$.

Let $x$ be a vertex of ${\mathfrak{X}}$; as above, we can assume $x=[X,{\rm
id}]$. We now define a map

$\Pi:\operatorname{lk}^{\downarrow}(x)\to\mathcal{P}(X,A),$

where $A$ is the set of all suited boundary components of $X$. Let
$z\in\operatorname{lk}^{\downarrow}(x)$. As such, we can write $z=[Z,g]$,
where there exists a surface $Z^{\prime}$ which is obtained from $Z$ by adding
pairwise disjoint pieces $Y_{0},\dots,Y_{p}$ so that $g$ maps $Z^{\prime}$ to
$X$ where $Z^{\prime}$ is a defining surface for $g$ as an $H$-block
homeomorphism. We then set $\Pi(z)=\\{g(Y_{0}),\dots,g(Y_{p})\\}$.

We first check that the map $\Pi$ is well-defined; to this end, let $(W,h)$ be
another representative of $z$. As before, this implies there exists a surface
$W^{\prime}$ obtained from $W$ by adding pairwise disjoint pieces
$Y_{0}^{\prime},\dots,Y_{p}^{\prime}$ so that $h$ maps $W^{\prime}$ to $X$. By
the definition of the equivalence relation, this means $h^{-1}\circ g$ maps
$Z$ to $W$ and in particular, $g(Z)=h(W)\subseteq X$. Therefore,
$\\{g(Y_{0}),\dots,g(Y_{p})\\}=\\{h(Y_{0}^{\prime}),\dots,h(Y_{p}^{\prime})\\}$,
and we are done.

Next, we have:

######  Proposition 6.10.

The map $\Pi$ is a complete join.

######  Remark 6.11.

In a previous version, we claimed that there was an isomorphism $\Pi$ from the
descending link to the _labeled block complex_. We are grateful to Chris
Leininger for pointing out that this was not correct, and for suggesting the
strategy for showing that it is a complete join.

###### Proof of Proposition 6.10.

Write $x=[X,\rm id]$, and let $\\{Z_{0},\ldots,Z_{p}\\}$ be a $p$-simplex in
$\mathcal{P}(X,A)$. Let $\\{Y_{0},\ldots,Y_{p}\\}$ be a collection of disjoint
blocks in $X$ such that each $Y_{i}$ has exactly two non-essential boundary
components. Now, $X\setminus\cup_{i=0}^{p}Z_{i}$ is homeomorphic to
$X\setminus\cup_{i=0}^{p}Y_{i}$ and thus there is a homeomorphism
$g:X\rightarrow X$ which takes $X\setminus\cup_{i=0}^{p}Y_{i}$ to
$X\setminus\cup_{i=0}^{p}Z_{i}$ and such that $g(Y_{i})=Z_{i}$. We conclude
that the map $\Pi$ is surjective.

Note that the disjointness condition on the links and on the block complex and
the fact that this property is naturally preserved by $\Pi$ implies that the
map is simplexwise injective.

It remains to show that the pre-image of a simplex is the join of the pre-
images of its vertices. Clearly the pre-image of a simplex is contained in the
join of the pre-images of its vertices so we only need to show the reverse
inclusion. To this end, let $\sigma=\\{Z_{0},\ldots,Z_{p}\\}$ be an
$p$-simplex in $\mathcal{P}(X,A)$. Take vertices

$[W_{0},g_{0}],\ldots,[W_{p},g_{p}]$

of $\mathfrak{X}$ such that $\Pi([W_{i},g_{i}])=Z_{i}$; we want to see that
these vertices in fact span an $p$-simplex in
$\operatorname{lk}^{\downarrow}(x)$.

We now make several reductions. We first claim that we may assume that, for
all $i=0,\ldots,p$, we have that $W_{i}\cup g_{i}^{-1}(Z_{i})=X$, so that
$g_{i}^{-1}(Z_{i})$ is a block and $g_{i}:X\rightarrow X$ is an element of
${\mathfrak{B}}$ with defining surface $X$. To see this, let
$Y_{i}=g_{i}^{-1}(Z_{i})$, which by construction is a block, and consider any
$H$-block homeomorphism $h$ which takes $W_{i}\cup Y_{i}$ to $X$ and such that
$W_{i}$ is a defining surface for $h$, i.e. the image of $Y_{i}$ is also a
block. Now we see that $(W_{i},g_{i})\sim(h(W_{i}),g_{i}\circ h^{-1})$, and
that this new representative has the desired properties.

We thus assume without loss of generality that $W_{i}\cup Y_{i}=X$ and
$g_{i}(X)=X$ for all $i=\ldots,p$. Next, we claim that we can assume that the
non-essential boundary components of $Y_{i}=g_{i}^{-1}(Z_{i})$ are disjoint
from those of $Y_{j}=g_{j}^{-1}(Z_{j})$ for all $i\neq j$. Indeed, given any
block $Y$ with two non-essential boundary components, there is an $H$-block
homeomorphism $f$ with $f(X)=X$ and $f(Y_{i})=Y$. Moreover
$(W_{i},g_{i})\sim(f(W_{i}),g_{i}\circ f^{-1})$. Since $f(Y_{i})=Y$, we can
freely choose the location of the block $Y_{i}$ in $X$ among blocks with two
non-essential boundary components.

We thus assume without loss of generality that $Y_{i}$ is disjoint from
$Y_{j}$ for each $i\neq j$. Finally, note that since $Y_{0},\dots,Y_{p}$ are
pairwise disjoint and $g_{i}(Y_{i})$ are pairwise disjoint and the complements
$X\backslash Y_{i}$ are homeomorphic to the complements $X\backslash
g_{i}(Y_{i})$, we can construct an element $g$ in ${\mathfrak{B}}$ with
$g(X)=X$ and which agrees with $g_{i}$ on $W_{i}$. In particular, the
collection $\\{[W_{i},g]\\}_{i=0}^{i=p}$ spans a simplex in
$\operatorname{lk}^{\downarrow}(x)$.

$\square$

######  Corollary 6.12 (Part “if” of Theorem 1.1).

If $H$ is of type $F_{n}$, so is $\mathfrak{B}_{d}(H)$.

###### Proof.

By the above results, since $H$ is of type $F_{n}$ then we can apply Brown’s
criterion above to the action of $\mathfrak{B}_{d}(H)$ on the cube complex
$\mathfrak{X}_{d}(O,Y,H)$, thus obtaining that $\mathfrak{B}_{d}(H)$ is of
type $F_{n}$, as desired. $\square$

##  7\. From global to local

Finally, in this section we prove that if $\mathfrak{B}_{d}(H)$ is of type
$F_{n}$, then so is $H$. In a nutshell, we will show that $H$ is a quasi-
retract of $\mathfrak{B}_{d}(H)$, following the strategy in [SW21, Section
3.5].

Before we proceed, we briefly recall the definition of quasi-retraction
between metric spaces. Let $X$ and $Y$ are metric spaces. A function $f:X\to
Y$ is said to be coarsely Lipschitz if there exist constants $C,D>0$ so that

$d(f(x),f(x^{\prime}))\leq Cd(x,x^{\prime})+D\text{ for all }x,x^{\prime}\in
X.$

A coarsely Lipschitz function $\rho:X\to Y$ is said to be a quasi-retraction
if there exists a coarse Lipschitz function $\iota:Y\to X$ and a constant
$E>0$ so that $d(\rho\circ\iota(y),y)\leq E$ for all $y\in Y$. In this
situation, we say that $Y$ is a quasi-retract of $X$. In the realm of groups,
we say that a finitely generated group $Q$ is a quasi-retract of another
finitely generated group $G$ if $Q$ is a quasi-retract of $G$ with respect to
the word metric given by some (and hence, any) choice of finite generating
sets. We have:

######  Theorem 7.1.

[Alo94, Theorem 8] Let $G$ and $Q$ be finitely generated groups such that $Q$
is a quasi-retract of $G$. If $G$ is of type $F_{n}$, so is $Q$.

After these definitions, observe that we have a canonical inclusion
homomorphism

$\iota:H\rightarrow\mathfrak{B}_{d}(H)$

given by the rule $h\mapsto[O_{2},({{\rm id}},(h,1,\ldots,1)),O_{2}]$, where
recall that $O_{2}$ is the surface that appears at the second step in the
construction of ${\mathcal{C}}_{d}(O,Y)$; in other words, $\iota(H)$ labels
the leftmost suited curve of $O_{2}$ by $h$ and the other suited curves by the
identity. Next, we define a map

$r:\mathfrak{B}_{d}(H)\to H$

by setting $r(f)=h_{1}$, where $f=[Z,(f_{Z},\mathbf{h}),f(Z)]$ and
$\mathbf{h}=(h_{1},\ldots,h_{n})$; observe that $r(f)$ does not depend on the
particular choice of such representative. We have:

######  Proposition 7.2.

$\mathfrak{B}_{d}(H)$ is generated by $\iota(H)$ and
$\mathfrak{B}_{d}=\mathfrak{B}_{d}(\\{1\\})$.

###### Proof.

Let $G$ be the group generated by $\iota(H)$ and $\mathfrak{B}_{d}$; we want
to prove that $G=\mathfrak{B}_{d}(H)$. To this end, let
$f\in\mathfrak{B}_{d}(H)$ and choose a tree-pair representative
$(Z,(f_{Z},\mathbf{h}),f(Z))$ for it, observing that

$[Z,(f_{Z},\mathbf{h}),f(Z)]=[Z,(f_{Z},{\rm id}),f(Z)]\cdot[f(Z),({\rm
id},\mathbf{h}),f(Z)]$

Since $[Z,(f_{Z},{\rm id}),f(Z)]\in\mathfrak{B}_{d}$, it suffices to show that
an element of the form $[X,({\rm id},\mathbf{h}),X]$ belongs to $G$. We can
assume that $O_{2}\subseteq X$.

Suppose for the time being that $X=O_{2}$. In this case, let
$\lambda_{j}(h):=(1,\cdots,1,h,1\cdots,1)$ where the only nontrivial labeling
$h$ appears on the $j$-th position, then

$[O_{2},({\rm id},\lambda_{j}(h)),O_{2}]=[O_{2},(g,{\rm
id}),O_{2}][O_{2},({\rm id},\lambda_{1}(h)),O_{2}][O_{2},(g^{-1},{\rm
id}),O_{2}],$

where $g\in\mathfrak{B}_{d}$ is any element supported on $O_{2}$ and that
interchanges the first and $j$-th suited boundary components of $O_{2}$,
fixing the rest. Since

$[O_{2},({\rm id},\lambda_{1}(h)),O_{2}]\in\iota(H)$

and for $\mathbf{h}=(h_{1}\cdots,h_{d})$

$[O_{2},({\rm id},\mathbf{h}),O_{2}]=\prod_{j=1}^{d}[O_{2},({\rm
id},\lambda_{j}(h_{j})),O_{2}],$

we are done in this case.

Suppose then that $X$ is a suited subsurface whose left most suited loop is
also a suited loop of $O_{2}$. By the same conjugation argument as in the
previous paragraph, we can assume that $\mathbf{h}=(h_{1},1,\ldots,1)$. But
then using the equivalence relation, we have

$[X,({\rm id},(h_{1},1,\ldots,1)),X]=[O_{2},({\rm
id},(h_{1},1,\ldots,1)),O_{2}],$

which already lies in the subgroup generated by $\iota(H)$ and
$\mathfrak{B}_{d}$. Now for an arbitrary suited subsurface $X$, again using
the conjugation argument, we can assume that $\mathbf{h}=(h_{1},1,\ldots,1)$.
But then $[X,({\rm id},\mathbf{h}_{1}),X]$ is conjugate to
$[X,(\mathrm{id},(1,\cdots,h_{1}),X]$ via an element in $\mathfrak{B}_{d}$.
Now we can find a suitable smaller suited subsurface $X^{\prime}$ whose left
most suited loop is also a suited loop of $O_{2}$, such that

$[X,(\mathrm{id},(1,\cdots,h_{1})),X]=[X^{\prime},(\mathrm{id},(1,\cdots,h_{1})),X^{\prime}].$

This is known to lie in the subgroup generated by $\iota(H)$ and
$\mathfrak{B}_{d}$ by the previous step and hence we are done. $\square$

######  Theorem 7.3.

When $H$ is finitely generated, it is a quasi-retract of
$\mathfrak{B}_{d}(H)$.

###### Proof.

Fix finite generating sets $S_{H}$ and $S_{B}$ for $H$ and $\mathfrak{B}_{d}$,
respectively. By Proposition 7.2, $\iota(S_{H})\cup S_{B}$ is a finite
generating set of $\mathfrak{B}_{d}(H)$. We will show that the map
$r:\mathfrak{B}_{d}(H)\rightarrow H$ described above is coarsely Lipschitz
with respect to the word metrics on $\mathfrak{B}_{d}(H)$ and $H$. To this
end:

  1. (1)

$r(g\iota(s))\in\\{r(g),r(g)s\\}$ for all $s\in S_{H}$ and all
$g\in\mathfrak{B}_{d}(H)$;

  2. (2)

$r(gg^{\prime})=r(g)$ for any $g^{\prime}\in\mathfrak{B}_{d}$ and
$g\in\mathfrak{B}_{d}(H)$.

In particular, $r$ is nonexpanding and hence coarsely Lipschitz. Since $\iota$
is a group homomorphism, it must be coarsely Lipschitz as well. As
$r\circ\iota={{\rm id}}_{H}$, we conclude that $r$ is a quasi-retraction.
$\square$

######  Corollary 7.4 (Part “only if” of Theorem 1.1).

If $\mathfrak{B}_{d}(H)$ is of type $F_{n}$, so is $H$.

###### Proof.

Let us first prove the corollary for $n=1$. If $H$ is not finitely generated,
we can find an increasing family of subgroups $H_{1}\lneq H_{2}\lneq
H_{n}\lneq\cdots$ of $H$ such that $\cup_{n=1}^{\infty}H_{i}=H$. But then we
would have a family of subgroups
$\mathfrak{B}_{d}(H_{1})\lneq\mathfrak{B}_{d}(H_{2})\lneq\mathfrak{B}_{d}(H_{n})\lneq\cdots$
of $\mathfrak{B}_{d}(H)$, such that
$\cup_{n=1}^{\infty}\mathfrak{B}_{d}(H_{i})=\mathfrak{B}_{d}(H)$. This implies
$\mathfrak{B}_{d}(H)$ is not finitely generated ,which is a contradiction.

Suppose now $\mathfrak{B}_{d}(H)$ is of type $F_{n}$ for some $n\geq 2$. As
above, $H$ is finitely generated, and the corollary now follows from Theorem
7.1 and Theorem 7.3.

$\square$

## References

  * [Alo94] Juan M. Alonso  “Finiteness conditions on groups and quasi-isometries”  In _J. Pure Appl. Algebra_ 95.2, 1994, pp. 121–129  DOI: 10.1016/0022-4049(94)90069-8
  * [AF21] Javier Aramayona and Louis Funar  “Asymptotic mapping class groups of closed surfaces punctured along Cantor sets”  In _Mosc. Math. J._ 21.1, 2021, pp. 1–29  DOI: 10.17323/1609-4514-2021-21-1-1-29
  * [AV20] Javier Aramayona and Nicholas G. Vlamis  “Big mapping class groups: an overview”  In _In the tradition of Thurston—geometry and topology_ Springer, Cham, 2020, pp. 459–496  DOI: 10.1007/978-3-030-55928-1“˙12
  * [Ara+21] Javier Aramayona et al.  “Generalized asymptotic mapping class groups and their finiteness properties” Preprint. arxiv: 2110.05318, 2021
  * [BB97] Mladen Bestvina and Noel Brady  “Morse theory and finiteness properties of groups”  In _Invent. Math._ 129.3, 1997, pp. 445–470  DOI: 10.1007/s002220050168
  * [Bri07] Matthew G. Brin  “The algebra of strand splitting. I. A braided version of Thompson’s group $V$”  In _J. Group Theory_ 10.6, 2007, pp. 757–788  DOI: 10.1515/JGT.2007.055
  * [Bro87] Kenneth S. Brown  “Finiteness properties of groups”  In _Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985)_ 44.1-3, 1987, pp. 45–75  DOI: 10.1016/0022-4049(87)90015-6
  * [BG84] Kenneth S. Brown and Ross Geoghegan  “An infinite-dimensional torsion-free ${\rm FP}_{\infty}$ group”  In _Invent. Math._ 77.2, 1984, pp. 367–381  DOI: 10.1007/BF01388451
  * [Bux+16] Kai-Uwe Bux et al.  “The braided Thompson’s groups are of type $\rm F_{\infty}$” With an appendix by Zaremsky  In _J. Reine Angew. Math._ 718, 2016, pp. 59–101  DOI: 10.1515/crelle-2014-0030
  * [Deh06] Patrick Dehornoy  “The group of parenthesized braids”  In _Adv. Math._ 205.2, 2006, pp. 354–409  DOI: 10.1016/j.aim.2005.07.012
  * [FM12] Benson Farb and Dan Margalit  “A primer on mapping class groups” 49, Princeton Mathematical Series  Princeton University Press, Princeton, NJ, 2012, pp. xiv+472
  * [Far03] Daniel S. Farley  “Finiteness and $\rm CAT(0)$ properties of diagram groups”  In _Topology_ 42.5, 2003, pp. 1065–1082  DOI: 10.1016/S0040-9383(02)00029-0
  * [FH15] Daniel S. Farley and Bruce Hughes  “Finiteness properties of some groups of local similarities”  In _Proc. Edinb. Math. Soc. (2)_ 58.2, 2015, pp. 379–402  DOI: 10.1017/S001309151400011X
  * [Flu+13] Martin G. Fluch, Marco Marschler, Stefan Witzel and Matthew C.. Zaremsky  “The Brin-Thompson groups $sV$ are of type $\text{F}_{\infty}$”  In _Pacific J. Math._ 266.2, 2013, pp. 283–295  DOI: 10.2140/pjm.2013.266.283
  * [FK04] L. Funar and C. Kapoudjian  “On a universal mapping class group of genus zero”  In _Geom. Funct. Anal._ 14.5, 2004, pp. 965–1012  DOI: 10.1007/s00039-004-0480-9
  * [FK09] Louis Funar and Christophe Kapoudjian  “An infinite genus mapping class group and stable cohomology”  In _Comm. Math. Phys._ 287.3, 2009, pp. 784–804  DOI: 10.1007/s00220-009-0728-1
  * [FK08] Louis Funar and Christophe Kapoudjian  “The braided Ptolemy-Thompson group is finitely presented”  In _Geom. Topol._ 12.1, 2008, pp. 475–530  DOI: 10.2140/gt.2008.12.475
  * [GLU20] Anthony Genevois, Anne Lonjou and Christian Urech  “Asymptotically rigid mapping class groups I: Finiteness properties of braided Thompson’s and Houghton’s groups” To appear. arxiv:2010.07225  In _Geom. Top._ , 2020
  * [Geo08] Ross Geoghegan  “Topological methods in group theory” 243, Graduate Texts in Mathematics  Springer, New York, 2008, pp. xiv+473  DOI: 10.1007/978-0-387-74614-2
  * [Har71] W.. Harvey  “On branch loci in Teichmüller space”  In _Trans. Amer. Math. Soc._ 153, 1971, pp. 387–399  DOI: 10.2307/1995564
  * [HW10] Allen Hatcher and Nathalie Wahl  “Stabilization for mapping class groups of 3-manifolds”  In _Duke Math. J._ 155.2, 2010, pp. 205–269  DOI: 10.1215/00127094-2010-055
  * [Ker23] B.. Kerékjártó  “Vorlesungen über Topologie. I”  Berlin: Spring, 1923
  * [Ner92] Yu.. Neretin  “Combinatorial analogues of the group of diffeomorphisms of the circle”  In _Izv. Ross. Akad. Nauk Ser. Mat._ 56.5, 1992, pp. 1072–1085  DOI: 10.1070/IM1993v041n02ABEH002264
  * [Ric63] Ian Richards  “On the classification of noncompact surfaces”  In _Trans. Amer. Math. Soc._ 106, 1963, pp. 259–269  DOI: 10.2307/1993768
  * [SW21] Rachel Skipper and Xiaolei Wu  “Finiteness properties for relatives of braided Higman–Thompson groups” Preprint. arxiv:2103.14589v2, 2021
  * [SW21a] Rachel Skipper and Xiaolei Wu  “Homological stability for the ribbon Higman–Thompson groups” Preprint. arxiv:2106.08751, 2021
  * [Ste92] Melanie Stein  “Groups of piecewise linear homeomorphisms”  In _Trans. Amer. Math. Soc._ 332.2, 1992, pp. 477–514  DOI: 10.2307/2154179
  * [Zar17] Matthew C.. Zaremsky  “Separation in the BNSR-invariants of the pure braid groups”  In _Publ. Mat._ 61.2, 2017, pp. 337–362  DOI: 10.5565/PUBLMAT6121702