Searcharxiv⌕ Search

arXiv subjects

Javier Aramayona

Publications and source records attributed to Javier Aramayona.

At least 19 recordsLinked to original sources
arXiv2026

Asymptotically conformal and asymptotically rigid mapping class groups

We give conditions ensuring that an asymptotically rigid mapping class group, specifically a surface Houghton group $\mathcal{H}(S)$, has finite index in the asymptotically conformal modular group $\text{Mod}_0(X)$, where $X$ is a hyperbolic structure on $S$. These include geometric conditions on the pieces of the underlying rigid structure, as well as the existence in $\mathcal{H}(S)$ of an end-periodic homeomorphism which is asymptotically conformal. As a consequence, if $S$ has $n\geq 3$ ends, then $\text{Mod}_0(X)$ has type $F_{n-1}$ but not $FP_n$. We also establish analogous results for $L^p$ modular groups.

math.GT↗
arXiv2026

Non-planar ends are continuously unforgettable

We show that continuous epimorphisms between a class of subgroups of mapping class groups of orientable infinite-genus 2-manifolds with no planar ends are always induced by homeomorphisms. This class of subgroups includes the pure mapping class group, the closure of the compactly supported mapping classes, and the full mapping class group in the case that the underlying manifold has a finite number of ends or is perfectly self-similar. As a corollary, these groups are Hopfian topological groups.

math.GT↗
arXiv2025

Cloning systems and action operads

Action operads and cloning systems are, respectively, the main ingredients in two approaches for axiomatically constructing Thompson-like groups due to Thumann and Witzel-Zaremsky. In this paper, we prove that action operads are equivalent to cloning systems that admit a certain extra structure, and which we call bilateral cloning systems. In addition, we describe their relation with crossed interval groups and product categories.

math.GR↗
arXiv2025

Hyperbolic spaces not quasi-isometric to curve complexes

We identify a condition that prevents a hyperbolic space from being quasi-isometric to the curve complex of any non-sporadic surface. Our result applies to several hyperbolic complexes, including arc complexes, disk complexes, non-separating curve complexes, (hyperbolic) pants complexes, and to free splitting complexes of free groups.

math.GT↗
arXiv2024

Quotients of the mapping class group by power subgroups

We study the quotient of the mapping class group $\operatorname{Mod}_g^n$ of a surface of genus $g$ with $n$ punctures, by the subgroup $\operatorname{Mod}_g^n[p]$ generated by the $p$-th powers of Dehn twists. Our first main result is that $\operatorname{Mod}_g^1 /\operatorname{Mod}_g^1[p]$ contains an infinite normal subgroup of infinite index, and in particular is not commensurable to a higher-rank lattice, for all but finitely many explicit values of $p$. Next, we prove that $\operatorname{Mod}_g^0/ \operatorname{Mod}_g^0[p]$ contains a Kähler subgroup of finite index, for every $p\ge 2$ coprime with six. Finally, we observe that the existence of finite-index subgroups of $\operatorname{Mod}_g^0$ with infinite abelianization is equivalent to the analogous problem for $\operatorname{Mod}_g^0/ \operatorname{Mod}_g^0[p]$.

math.GT↗
arXiv2024

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

We prove that the infinite family of asymptotic mapping class groups of surfaces of defined by Funar--Kapoudjian and Aramayona--Funar are of type $F_\infty$, thus answering questions of Funar-Kapoudjian-Sergiescu and Aramayona-Vlamis. As it turns out, this result is a specific instance of a much more general theorem which allows to deduce that asymptotic mapping class groups of Cantor manifolds, also introduced in this paper, are of type $F_\infty$, provide the underlying manifolds satisfy some general hypotheses. As important examples, we will obtain $F_\infty$ asymptotical mapping class groups that contain, respectively, the mapping class group of every compact surface with non-empty boundary, the automorphism group of every free group of finite rank, or infinite families of arithmetic groups. In addition, for certain types of manifolds, the homology of our asymptotic mapping class groups coincides with the stable homology of the relevant mapping class groups, as studied by Harer and Hatcher--Wahl.

math.GT↗
arXiv2023

Block mapping class groups and their finiteness properties

A Cantor surface $\mathcal C_d$ is a non-compact surface obtained by gluing copies of a fixed compact surface $Y^d$ (a block), with $d+1$ boundary components, in a tree-like fashion. For a fixed subgroup $H<Map(Y^d)$ , we consider the subgroup $\mathfrak B_d(H)<Map(\mathcal C_d)$ whose elements eventually send blocks to blocks and act like an element of $H$; we refer to $\mathfrak B_d(H)$ as the block mapping class group with local action prescribed by $H$. The family of groups so obtained contains the asymptotic mapping class groups of \cite{SW21a,ABF+21, FK04}. Moreover, there is a natural surjection onto the family symmetric Thompson groups of Farley--Hughes \cite{FH15}; in particular, they provide a positive answer to \cite[Question 5.37]{AV20}. We prove that, when the block is a (holed) sphere or a (holed) torus, $\mathfrak B_d(H)$ is of type $F_n$ if and only if $H$ is of type $F_n$. As a consequence, for every $n$, $Map(C_d)$ has a subgroup of type $F_n$ but not $F_{n+1}$ which contains the mapping class group of every compact subsurface of $\mathcal C_d$.

math.GT↗
arXiv2023

Surface Houghton groups

For every $n\ge 2$, the {\em surface Houghton group} $\mathcal B_n$ is defined as the asymptotically rigid mapping class group of a surface with exactly $n$ ends, all of them non-planar. The groups $\mathcal B_n$ are analogous to, and in fact contain, the braided Houghton groups. These groups also arise naturally in topology: every monodromy homeomorphisms of a fibered component of a depth-1 foliation of closed 3-manifold is conjugate into some $\mathcal B_n$. As countable mapping class groups of infinite type surfaces, the groups $\mathcal B_n$ lie somewhere between classical mapping class groups and big mapping class groups. We initiate the study of surface Houghton groups proving, among other things, that $\mathcal B_n$ is of type $F_{n-1}$, but not of type $FP_n$, analogous to the braided Houghton groups.

math.GT↗
arXiv2021

On the geometry of graphs associated to infinite-type surfaces

Consider a connected orientable surface $S$ of infinite topological type, i.e. with infinitely-generated fundamental group. We describe the large-scale geometry of arbitrary connected subgraphs of the arc complex $A(S)$ and curve complex $C(S)$ of $S$, provided they are invariant under a sufficiently big subgroup of the mapping class group $Mod(S)$. We obtain a number of consequences; in particular we recover the main results of J. Bavard and Aramayona-Fossas-Parlier .

math.GT↗
arXiv2021

Big mapping class groups and the co-Hopfian property

We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are the first examples of injective endomorphisms of mapping class groups (of surfaces with empty boundary) that fail to be surjective. We then prove that, subject to some topological conditions on the domain surface, any continuous injective homomorphism between (arbitrary) big mapping class groups that sends Dehn twists to Dehn twists is induced by homeomorphism. Finally, we explore the extent to which, in stark contrast to the finite-type case, superinjective maps between curve graphs impose no topological restrictions on the underlying surfaces.

math.GT↗
arXiv2020

The first integral cohomology of pure mapping class groups

It is a classical result of Powell that pure mapping class groups of connected, orientable surfaces of finite type and genus at least three are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surface's simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.

math.GT↗
arXiv2019

Asymptotic mapping class groups of closed surfaces punctured along Cantor sets

We introduce subgroups ${\mathcal{B}}_g< {\mathcal H}_g$ of the mapping class group $Mod(Σ_g)$ of a closed surface of genus $g \ge 0$ with a Cantor set removed, which are extensions of Thompson's group $V$ by a direct limit of mapping class groups of compact surfaces of genus $g$. We first show that both ${\mathcal{B}}_g$ and ${\mathcal H}_g$ are finitely presented, and that ${\mathcal H}_g$ is dense in $Mod(Σ_g)$. We then exploit the relation with Thompson's groups to study properties ${\mathcal B}_g$ and ${\mathcal H}_g$ in analogy with known facts about finite-type mapping class groups. For instance, their homology coincides with the stable homology of the mapping class group of genus $g$, every automorphism is geometric, and every homomorphism from a higher-rank lattice has finite image. In addition, the same connection with Thompson's groups will also prove that ${\mathcal B}_g$ and ${\mathcal H}_g$ are not linear and do not have Kazhdan's Property (T), which represents a departure from the current knowledge about finite-type mapping class groups.

math.GT↗
arXiv2019

Big Torelli groups: generation and commensuration

For any surface $Σ$ of infinite topological type, we study the Torelli subgroup ${\mathcal I}(Σ)$ of the mapping class group ${\rm MCG}(Σ)$, whose elements are those mapping classes that act trivially on the homology of $Σ$. Our first result asserts that ${\mathcal I}(Σ)$ is topologically generated by the subgroup of ${\rm MCG}(Σ)$ consisting of those elements in the Torelli group which have compact support. In particular, using results of Birman, Powell, and Putman we deduce that ${\mathcal I}(Σ)$ is topologically generated by separating twists and bounding pair maps. Next, we prove the abstract commensurator group of ${\mathcal I}(Σ)$ coincides with ${\rm MCG}(Σ)$. This extends the results for finite-type surfaces of Farb-Ivanov, Brendle-Margalit and KIda to the setting of infinite-type surfaces.

math.GT↗
arXiv2018

A note on nilpotent subgroups of automorphism groups of RAAGs

We observe that automorphism groups of right-angled Artin groups contain nilpotent non-abelian subgroups, namely $H_3(\mathbb{Z})$ the three-dimensional integer Heisenberg group, provided they admit a certain type of element, called an adjacent transvection. This represents a (minor) extension of a result of Charney-Vogtmann.

math.GR↗