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Matthew C. B. Zaremsky

Publications and source records attributed to Matthew C. B. Zaremsky.

At least 37 records · Page 2Linked to original sources
arXiv2023

Braiding groups of automorphisms and almost-automorphisms of trees

We introduce "braided" versions of self-similar groups and Röver--Nekrashevych groups, and study their finiteness properties. This generalizes work of Aroca and Cumplido, and the first author and Wu, who considered the case when the self-similar groups are what we call "self-identical". In particular we use a braided version of the Grigorchuk group to construct a new group called the braided Röver group, which we prove is of type $F_\infty$. Our techniques involve using so called $d$-ary cloning systems to construct the groups, and analyzing certain complexes of embedded disks in a surface to understand their finiteness properties.

math.GR↗
arXiv2022

Random subcomplexes of finite buildings, and fibering of commutator subgroups of right-angled Coxeter groups

The main theme of this paper is higher virtual algebraic fibering properties of right-angled Coxeter groups (RACGs), with a special focus on those whose defining flag complex is a finite building. We prove for particular classes of finite buildings that their random induced subcomplexes have a number of strong properties, most prominently that they are highly connected. From this we are able to deduce that the commutator subgroup of a RACG, with defining flag complex a finite building of a certain type, admits an epimorphism to $\mathbb{Z}$ whose kernel has strong topological finiteness properties. We additionally use our techniques to present examples where the kernel is of type $\textrm{F}_2$ but not $\textrm{FP}_3$, and examples where the RACG is hyperbolic and the kernel is finitely generated and non-hyperbolic. The key tool we use is a generalization of an approach due to Jankiewicz-Norin-Wise involving Bestvina-Brady discrete Morse theory applied to the Davis complex of a RACG, together with some probabilistic arguments.

math.GR↗
arXiv2022

Braided Thompson groups with and without quasimorphisms

We study quasimorphisms and bounded cohomology of a variety of braided versions of Thompson groups. Our first main result is that the Brin--Dehornoy braided Thompson group $bV$ has an infinite-dimensional space of quasimorphisms and thus infinite-dimensional second bounded cohomology. This implies that despite being perfect, $bV$ is not uniformly perfect, in contrast to Thompson's group $V$. We also prove that relatives of $bV$ like the ribbon braided Thompson group $rV$ and the pure braided Thompson group $bF$ similarly have an infinite-dimensional space of quasimorphisms. Our second main result is that, in stark contrast, the close relative of $bV$ denoted $\hat{bV}$, which was introduced concurrently by Brin, has trivial second bounded cohomology. This makes $\hat{bV}$ the first example of a left-orderable group of type $\operatorname{F}_\infty$ that is not locally indicable and has trivial second bounded cohomology. This also makes $\hat{bV}$ an interesting example of a subgroup of the mapping class group of the plane minus a Cantor set that is non-amenable but has trivial second bounded cohomology, behaviour that cannot happen for finite-type mapping class groups.

math.GR↗
arXiv2022

A user's guide to cloning systems

In joint work of the author with Stefan Witzel, a procedure was developed for building new examples of groups in the extended family of R. Thompson's groups, using what we termed \emph{cloning systems}. These new Thompson-like groups can be thought of as limits of families of groups, though unlike other limiting processes, e.g., direct limits, these tend to be well behaved with respect to finiteness properties. In this expository note, we distill the crucial parts of that 50-page paper into a more digestible form, for those curious to understand the construction but less curious about the gritty details. We also give one new example, of a cloning system involving signed symmetric groups.

math.GR↗
arXiv2022

Von Neumann algebras of Thompson-like groups from cloning systems

We prove a variety of results about the group von Neumann algebras associated to Thompson-like groups arising from so called $d$-ary cloning systems. Cloning systems are a framework developed by Witzel and the second author, with a $d$-ary version subsequently developed by Skipper and the second author, which can be used to construct generalizations of the classical Thompson's groups $F$, $T$, and $V$. Given a family of groups $(G_n)_{n\in\mathbb{N}}$ with a $d$-ary cloning system, we get a Thompson-like group $\mathscr{T}_d(G_*)$, and in this paper we find some mild, natural conditions under which the group von Neumann algebra $\mathcal{L}(\mathscr{T}_d(G_*))$ has desirable properties. For instance, if the $d$-ary cloning system is "fully compatible" and "diverse" then we prove that $\mathcal{L}(\mathscr{T}_d(G_*))$ is a type $\textrm{II}_1$ factor. If moreover the $d$-ary cloning system is "uniform" and "slightly pure" then we prove $\mathcal{L}(\mathscr{T}_d(G_*))$ is even a McDuff factor, so $\mathscr{T}_d(G_*)$ is inner amenable. Examples of $d$-ary cloning systems satisfying these conditions are easy to come by, and include many existing examples, for instance our results show that for $bV$ and $bF$ the Brin-Dehornoy braided Thompson group and pure braided Thompson group, $\mathcal{L}(bV)$ and $\mathcal{L}(bF)$ are type $\textrm{II}_1$ factors and $\mathcal{L}(bF)$ is McDuff. In particular we get the surprising result that $bF$ is inner amenable.

math.OA↗
arXiv2022

A taste of twisted Brin-Thompson groups

This note serves as a short and reader-friendly introduction to twisted Brin-Thompson groups, which were recently constructed by Belk and the author to provide a family of simple groups with a variety of interesting properties. Most notably, twisted Brin-Thompson groups can be used to show that every finitely generated group quasi-isometrically embeds as a subgroup of a finitely generated simple group. Another important application is a concrete construction of a family of simple groups with arbitrary finiteness length. In addition to giving a concise introduction to the groups and these applications, we also prove here a strengthening of one of the results from the original paper. Namely, we prove that any finitely presented group acting faithfully and oligomorphically on a set, with finitely generated stabilizers of finite subsets, embeds in a finitely presented simple group. We believe this could potentially lead to future progress on resolving the Boone-Higman Conjecture for certain groups.

math.GR↗
arXiv2021

Higher connectivity of the Morse complex

The Morse complex $\mathcal{M}(Δ)$ of a finite simplicial complex $Δ$ is the complex of all gradient vector fields on $Δ$. In this paper we study higher connectivity properties of $\mathcal{M}(Δ)$. For example, we prove that $\mathcal{M}(Δ)$ gets arbitrarily highly connected as the maximum degree of a vertex of $Δ$ goes to $\infty$, and for $Δ$ a graph additionally as the number of edges goes to $\infty$. We also classify precisely when $\mathcal{M}(Δ)$ is connected or simply connected. Our main tool is Bestvina-Brady Morse theory, applied to a "generalized Morse complex."

math.CO↗
arXiv2021

The braided Thompson's groups are of type $F_\infty$

We prove that the braided Thompson's groups $V_{\rm br}$ and $F_{\rm br}$ are of type $F_\infty$, confirming a conjecture by John Meier. The proof involves showing that matching complexes of arcs on surfaces are highly connected. In an appendix, Zaremsky uses these connectivity results to exhibit families of subgroups of the pure braid group that are highly generating, in the sense of Abels and Holz.

math.GR↗
arXiv2021

Twisted Brin-Thompson groups

We construct a family of infinite simple groups that we call \emph{twisted Brin-Thompson groups}, generalizing Brin's higher-dimensional Thompson groups $sV$ ($s\in\mathbb{N}$). We use twisted Brin-Thompson groups to prove a variety of results regarding simple groups. For example, we prove that every finitely generated group embeds quasi-isometrically as a subgroup of a two-generated simple group, strengthening a result of Bridson. We also produce examples of simple groups that contain every $sV$ and hence every right-angled Artin group, including examples of type $\textrm{F}_\infty$ and a family of examples of type $\textrm{F}_{n-1}$ but not of type $\textrm{F}_n$, for arbitrary $n\in\mathbb{N}$. This provides the second known infinite family of simple groups distinguished by their finiteness properties.

math.GR↗
arXiv2021

Almost-automorphisms of trees, cloning systems and finiteness properties

We prove that the group of almost-automorphisms of the infinite rooted regular $d$-ary tree $\mathcal{T}_d$ arises naturally as the Thompson-like group of a so called $d$-ary cloning system. A similar phenomenon occurs for any Röver-Nekrashevych group $V_d(G)$, for $G\le Aut(\mathcal{T}_d)$ a self-similar group. We use this framework to expand on work of Belk and Matucci, who proved that the Röver group, using the Grigorchuk group for $G$, is of type $F_\infty$. Namely, we find some natural conditions on subgroups of $G$ to ensure that $V_d(G)$ is of type $F_\infty$, and in particular we prove this for all $G$ in the infinite family of Šunić groups. We also prove that if $G$ is itself of type $F_\infty$ then so is $V_d(G)$, and that every finitely generated virtually free group is self-similar, so in particular every finitely generated virtually free group $G$ yields a type $F_\infty$ Röver-Nekrashevych group $V_d(G)$.

math.GR↗
arXiv2021

Geometric structures related to the braided Thompson groups

In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type $\textrm{F}_\infty$. The proof utilized certain contractible cube complexes, which in this paper we prove are CAT(0). We then use this fact to compute the geometric invariants $Σ^m(F_{\textrm{br}})$ of the pure braided Thompson group $F_{\textrm{br}}$. Only the first invariant $Σ^1(F_{\textrm{br}})$ was previously known. A consequence of our computation is that as soon as a subgroup of $F_{\textrm{br}}$ containing the commutator subgroup $[F_{\textrm{br}},F_{\textrm{br}}]$ is finitely presented, it is automatically of type $\textrm{F}_\infty$.

math.GR↗
arXiv2021

Bestvina-Brady discrete Morse theory and Vietoris-Rips complexes

We inspect Vietoris-Rips complexes $VR_t(X)$ of certain metric spaces $X$ using a new generalization of Bestvina-Brady discrete Morse theory. Our main result is a pair of metric criteria on $X$, called the Morse Criterion and Link Criterion, that allow us to deduce information about the homotopy types of certain $VR_t(X)$. One application is to topological data analysis, specifically persistence of homotopy type for certain Vietoris-Rips complexes. For example we recover some results of Adamaszek-Adams and Hausmann regarding homotopy types of $VR_t(S^n)$. Another application is to geometric group theory; we prove that any group acting geometrically on a metric space satisfying a version of the Link Criterion admits a geometric action on a contractible simplicial complex, which has implications for the finiteness properties of the group. This applies for example to asymptotically $CAT(0)$ groups. We also prove that any group with a word metric satisfying the Link Criterion in an appropriate range has a contractible Vietoris-Rips complex, and use combings to exhibit a family of groups with this property.

math.GT↗
arXiv2020

The BNSR-invariants of the Stein group $F_{2,3}$

The Stein group $F_{2,3}$ is the group of orientation-preserving homeomorphisms of the unit interval with slopes of the form $2^p3^q$ ($p,q\in\mathbb{Z}$) and breakpoints in $\mathbb{Z}[\frac{1}{6}]$. This is a natural relative of Thompson's group $F$. In this paper we compute the Bieri-Neumann-Strebel-Renz (BNSR) invariants $Σ^m(F_{2,3})$ of the Stein group for all $m\in\mathbb{N}$. A consequence of our computation is that (as with $F$) every finitely presented normal subgroup of $F_{2,3}$ is of type $\textrm{F}_\infty$. Another, more surprising, consequence is that (unlike $F$) the kernel of any map $F_{2,3}\to\mathbb{Z}$ is of type $\textrm{F}_\infty$, even though there exist maps $F_{2,3}\to \mathbb{Z}^2$ whose kernels are not even finitely generated. In terms of BNSR-invariants, this means that every discrete character lies in $Σ^\infty(F_{2,3})$, but there exist (non-discrete) characters that do not even lie in $Σ^1(F_{2,3})$. To the best of our knowledge, $F_{2,3}$ is the first group whose BNSR-invariants are known exhibiting these properties.

math.GR↗
arXiv2020

The BNSR-invariants of the Lodha-Moore groups, and an exotic simple group of type $\textrm{F}_\infty$

In this paper we give a complete description of the Bieri-Neumann-Strebel-Renz invariants of the Lodha-Moore groups. The second author previously computed the first two invariants, and here we show that all the higher invariants coincide with the second one, which finishes the complete computation. As a consequence, we present a complete picture of the finiteness properties of normal subgroups of the first Lodha-Moore group. In particular, we show that every finitely presented normal subgroup of the group is of type $\textrm{F}_\infty$, answering question 112 from Oberwolfach Rep., 15(2):1579-1633, 2018. The proof involves applying a variation of Bestvina-Brady discrete Morse theory to the so called cluster complex $X$ introduced by the first author. As an application, we also demonstrate that a certain simple group $S$ previously constructed by the first author is of type $\textrm{F}_\infty$. This provides the first example of a type $\textrm{F}_\infty$ simple group that acts faithfully on the circle by homeomorphisms, but does not admit any nontrivial action by $C^1$-diffeomorphisms, nor by piecewise linear homeomorphisms, on any $1$-manifold.

math.GR↗
arXiv2020

A short account of why Thompson's group $F$ is of type $\textrm{F}_\infty$

In 1984 Brown and Geoghegan proved that Thompson's group $F$ is of type $\textrm{F}_\infty$, making it the first example of an infinite dimensional torsion-free group of type $\textrm{F}_\infty$. Over the decades a different, shorter proof has emerged, which is more streamlined and generalizable to other groups. It is difficult, however, to isolate this proof in the literature just for $F$ itself, with no complicated generalizations considered and no additional properties proved. The goal of this expository note then is to present the "modern" proof that $F$ is of type $\textrm{F}_\infty$, and nothing else.

math.GR↗
arXiv2019

The BNSR-invariants of the Houghton groups, concluded

We give a complete computation of the BNSR-invariants $Σ^m(H_n)$ of the Houghton groups $H_n$. Partial results were previously obtained by the author, with a conjecture about the full picture, which we now confirm. The proof involves covering relevant subcomplexes of an associated $CAT(0)$ cube complex by their intersections with certain locally convex subcomplexes, and then applying a strong form of the Nerve Lemma. A consequence of the full computation is that for each $1\le m\le n-1$, $H_n$ admits a map onto $\mathbb{Z}$ whose kernel is of type $F_{m-1}$ but not $F_m$, and moreover no such kernel is ever of type $F_{n-1}$.

math.GR↗
arXiv2018

Thompson groups for systems of groups, and their finiteness properties

We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pure braid groups. We give new examples of families of groups that admit a cloning system and study how the finiteness properties of the resulting generalized Thompson group depend on those of the original groups. The main new examples here include upper triangular matrix groups, mock reflection groups, and loop braid groups. For generalized Thompson groups of upper triangular matrix groups over rings of S-integers of global function fields, we develop new methods for (dis-)proving finiteness properties, and show that the finiteness length of the generalized Thompson group is exactly the limit inferior of the finiteness lengths of the groups in the family.

math.GR↗