arXiv subjects

Matthew C. B. Zaremsky

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

At least 19 recordsLinked to original sources

arXiv2026

Boone-Higman embeddings of $\mathrm{Aut}(F_n)$ and mapping class groups of punctured surfaces

We prove that the groups $\mathrm{Aut}(F_n)$ satisfy the Boone-Higman conjecture for all $n$, meaning each $\mathrm{Aut}(F_n)$ embeds in a finitely presented simple group. In fact, we prove that each $\mathrm{Aut}(F_n)$ satisfies the "permutational" Boone-Higman conjecture, which means the simple group in question can be taken to be a twisted Brin-Thompson group. A far-reaching consequence of our approach is that finitely presented twisted Brin-Thompson groups are universal among finitely presented simple groups that are highly transitive. This is evidence toward the Boone-Higman conjecture being equivalent to its permutational version. Proving the conjecture for $\mathrm{Aut}(F_n)$ also confirms the conjecture for all groups (virtually) embedding into some $\mathrm{Aut}(F_n)$, such as mapping class groups of non-closed surfaces, braid groups, loop braid groups, ribbon braid groups and certain Artin groups. This answers several questions of the first and fourth authors with Bleak and Matucci. Yet another consequence of our approach is that satisfying the permutational Boone-Higman conjecture is closed under free products and regular wreath products, which answers a question of the fourth author, confirms the conjecture for free solvable groups, and reduces the conjecture for finitely generated $3$-manifold groups to the case of non-Seifert fibered closed graph manifolds.

math.GR↗

arXiv2026

Word length, Morse theory, and Vietoris-Rips complexes

We present a discrete Morse theoretic approach to proving high connectivity or contractibility of a Vietoris-Rips complex using distance to a fixed point as an initial measurement. In particular we focus on the case of a finitely generated group using word length. As a proof-of-concept application we prove that $\mathcal{VR}_2(A_Γ)$ is contractible for $A_Γ$ a right-angled Artin group on a triangle-free graph $Γ$. We also prove an interesting sufficient condition for a group to be finitely presented that only requires checking connectivity of certain finite complexes.

math.GR↗

arXiv2026

Finiteness properties and Higman's rope trick

In 1961 Higman proved that every finitely generated recursively presented group embeds into a finitely presented group. In 2018 Leary proved that every finitely generated group embeds into a group of type $FP_2$. One naturally wonders whether analogous results hold for the higher finiteness properties $F_n$ and $FP_n$ ($3 \leq n \leq \infty$), i.e., whether every finitely generated recursively presented group embeds into a group of type $F_n$ and whether every finitely generated group embeds into a group of type $FP_n$. We prove that a positive answer to the $FP_n$ question that moreover preserves recursive presentability would imply a positive answer to the $F_n$ question. We also investigate the output groups from Higman's and Leary's proofs, which in both cases arise from the so-called ``Higman rope trick'', and find that they are never of type $FP_3(\mathbb{Q})$ (hence also never $FP_3$ nor $F_3$). Thus, any approach to the questions of higher finiteness properties must go via a different route than the Higman rope trick.

math.GR↗

arXiv2026

Action graphs, semiconjugacy, and non-embedding in Thompson's group $V$

We prove a variety of results about subgroups of Thompson's group $V$. First we prove that every action graph of a finitely generated subgroup of $V$ acting on an orbit in Cantor space is quasi-isometric to a tree. Then we prove that for a broad class of groups of homeomorphisms of the real line, for example Thompson's group $F$, any action on the Cantor space via an embedding into Thompson's group $V$ must be semiconjugate to the standard action on the line. Finally, we use this to establish that many such groups cannot embed into $V$; in particular the Stein group $F_{2,3}$ cannot embed in $V$, answering a question of the third author.

math.GR↗

arXiv2026

Hyperbolic actions of Thompson's group $F$ and generalizations

We study the poset of hyperbolic structures on Thompson's group $F$ and its generalizations $F_n$ for $n \geq 2$. The global structure of this poset is as simple as one would expect, with the maximal non-elementary elements being two quasi-parabolic actions corresponding to well-known ascending HNN-extension expressions of $F_n$. However, the local structure turns out to be incredibly rich, in stark contrast with the situation for the $T$ and $V$ counterparts. We show that the subposet of quasi-parabolic hyperbolic structures consists of two isomorphic posets, each of which contains uncountably many subposets of \emph{lamplike} structures, which can be described combinatorially in terms of certain hyperbolic structures on related lamplighter groups. Moreover, each of these subposets, as well as intersections and complements thereof, is very large, in that it contains a copy of the power set of the natural numbers. We also prove that these uncountably many uncountable subposets are not the entire picture, indeed there exists a copy of the power set of the natural numbers consisting entirely of non-lamplike structures. We also prove that this entire vast array of hyperbolic structures on $F_n$ collapses as soon as one takes a natural semidirect product with $\mathbb{Z}/2\mathbb{Z}$. These results are all proved via a detailed analysis of confining subsets, and along the way we establish a number of fundamental results in the theory of confining subsets of groups.

math.GR↗

arXiv2026

Abstract twisted Brin--Thompson groups

Given a group $G$ acting faithfully on a set $S$, one gets a simple group denoted $SV_G$, called a twisted Brin--Thompson group. In this paper we drop the faithfulness assumption, and get an abstract version of a twisted Brin--Thompson group $SV_G$. While the resulting group is not simple, since $SV_G$ surjects onto $SV_{G/\ker(G \curvearrowright S)}$, we prove that every proper normal subgroup of $SV_G$ lies in the kernel of this surjection, so $SV_G$ is ``relatively simple''. The advantage is that now we can prove that every finitely presented simple group embeds in a finitely presented abstract twisted Brin--Thompson group intersecting this kernel trivially. In particular, if the Boone--Higman conjecture is true, then so is a related conjectural characterization of groups with solvable word problem, arising purely in the world of twisted Brin--Thompson groups. We also prove a variety of additional results about abstract twisted Brin--Thompson groups, some of which are new even in the faithful case: they are all uniformly perfect, have property NL and property FW$_\infty$, are boundedly acyclic and $\ell^2$-invisible, and are $C^*$-simple as soon as they have trivial amenable radical. Along the way we formulate a new general criterion for $\ell^2$-invisibility that is interesting in its own right.

math.GR↗

arXiv2026

Finiteness properties of stabilisers of oligomorphic actions

An action of a group on a set is oligomorphic if it has finitely many orbits of $n$-element subsets for all $n$. We prove that for a large class of groups (including all groups of finite virtual cohomological dimension and all countable linear groups), for any oligomorphic action of such a group on an infinite set there exists a finite subset whose stabiliser is not of type $\mathrm{FP}_\infty$. This leads to obstructions on finiteness properties for permutational wreath products and twisted Brin-Thompson groups. We also prove a version for actions on flag complexes, and discuss connections to the Boone-Higman conjecture. In the appendix, we improve on the criterion of Bartholdi-Cornulier-Kochloukova for finiteness properties of wreath products, and the criterion of Kropholler-Martino for finiteness properties of graph-wreath products.

math.GR↗

arXiv2026

Dense and empty BNSR-invariants of the McCool groups

An automorphism of the free group $F_n$ is called pure symmetric if it sends each generator to a conjugate of itself. The group $\mathrm{PSA}_n$ of all pure symmetric automorphisms and its quotient $\mathrm{PSO}_n$ by the group of inner automorphisms are called the McCool groups. In this paper we prove that every BNSR-invariant $Σ^m$ of a McCool group is either dense or empty in the character sphere, and we characterize precisely when each situation occurs. Our techniques involve understanding higher generation properties of abelian subgroups of McCool groups, coming from the McCullough-Miller space. We also investigate further properties of the second invariant $Σ^2$ for McCool groups using a general criterion due to Meinert for a character to lie in $Σ^2$.

math.GR↗

arXiv2026

Configuration spaces of circles in the plane

We consider the space of all configurations of finitely many (potentially nested) circles in the plane. We prove that this space is aspherical, and compute the fundamental group of each of its connected components. It turns out these fundamental groups are obtained as iterated semidirect products of subgroups of braid groups, with the structure for each component dictated by a finite rooted tree. These groups can be viewed as "braided" versions of the automorphism groups of such trees. We also discuss connections to statistical mechanics, topological data analysis, and geometric group theory.

math.AT↗

arXiv2025

Hyperbolic groups satisfy the Boone-Higman conjecture

The 1973 Boone-Higman conjecture predicts that every finitely generated group with solvable word problem embeds in a finitely presented simple group. In this paper, we show that hyperbolic groups satisfy this conjecture, that is, each hyperbolic group embeds in some finitely presented simple group. This shows that the conjecture holds in the "generic" case for finitely presented groups. Our key tool is a new family of groups, which we call "rational similarity groups (RSGs)", that is interesting in its own right. We prove that every hyperbolic group embeds in a full, contracting RSG, and every full, contracting RSG embeds in a finitely presented simple group, thus establishing the result. Another consequence of our work is that all contracting self-similar groups satisfy the Boone-Higman conjecture.

math.GR↗

arXiv2025

Contractible Vietoris-Rips complexes of $\mathbb{Z}^n$

We give a new, short proof of a result of Virk, that the Vietoris-Rips complex of the group $\mathbb{Z}^n$ with the standard word metric is contractible at large enough scales. This is inspired by a key observation in Virk's proof, but we use Bestvina-Brady discrete Morse theory to get a very short proof with better bounds. In the course of this, we get a new, general criterion for a metric space to have contractible Vietoris-Rips complexes at large enough scales, which could prove useful in the future.

math.GR↗

arXiv2025

Non-inner amenability of the Higman-Thompson groups

We prove that the Higman-Thompson groups $T_n$ and $V_n$ are non-inner amenable for all $n\ge 2$. This extends Haagerup and Olesen's result that Thompson's groups $T=T_2$ and $V=V_2$ are non-inner amenable. Their proof relied on machinery only available in the $n=2$ case, namely Thurston's piecewise-projective model for Thompson's group $T$, so our approach necessarily utilizes different tools. This also provides an alternate proof of Haagerup-Olesen's result when $n=2$.

math.GR↗

arXiv2025

Progress around the Boone-Higman Conjecture

A conjecture of Boone and Higman from the 1970's asserts that a finitely generated group $G$ has solvable word problem if and only if $G$ can be embedded into a finitely presented simple group. We comment on the history of this conjecture and survey recent results that establish the conjecture for many large classes of interesting groups.

math.GR↗

arXiv2025

Embedding finitely presented self-similar groups into finitely presented simple groups

We prove that every finitely presented self-similar group embeds in a finitely presented simple group. This establishes that every group embedding in a finitely presented self-similar group satisfies the Boone-Higman conjecture. The simple groups in question are certain commutator subgroups of Röver-Nekrashevych groups, and the difficulty lies in the fact that even if a Röver-Nekrashevych group is finitely presented, its commutator subgroup might not be. We also discuss a general example involving matrix groups over certain rings, which in particular establishes that every finitely generated subgroup of $\mathrm{GL}_n(\mathbb{Q})$ satisfies the Boone-Higman conjecture.

math.GR↗

arXiv2024

Finite presentability of twisted Brin-Thompson groups

Given a group $G$ acting faithfully on a set $S$, we characterize precisely when the twisted Brin-Thompson group $SV_G$ is finitely presented. The answer is that $SV_G$ is finitely presented if and only if we have the following: $G$ is finitely presented, the action of $G$ on $S$ has finitely many orbits of two-element subsets of $S$, and the stabilizer in $G$ of any element of $S$ is finitely generated. Since twisted Brin-Thompson groups are simple, a consequence is that any subgroup of a group admitting an action as above satisfies the Boone-Higman conjecture. In the course of proving this, we also establish a sufficient condition for a group acting cocompactly on a simply connected complex to be finitely presented, even if certain edge stabilizers are not finitely generated, which may be of independent interest.

math.GR↗

arXiv2024

Houghton-like groups from "shift-similar" groups

We introduce and study \emph{shift-similar} groups $G\le\textrm{Sym}(\mathbb{N})$, which play an analogous role in the world of Houghton groups that self-similar groups play in the world of Thompson groups. We also introduce Houghton-like groups $H_n(G)$ arising from shift-similar groups $G$, which are an analog of Röver-Nekrashevych groups from the world of Thompson groups. We prove a variety of results about shift-similar groups and these Houghton-like groups, including results about finite generation and amenability. One prominent result is that every finitely generated group embeds as a subgroup of a finitely generated shift-similar group, in contrast to self-similar groups, where this is not the case. This establishes in particular that there exist uncountably many isomorphism classes of finitely generated shift-similar groups, again in contrast to the self-similar situation.

math.GR↗

arXiv2024

Finitely presented simple groups with at least exponential Dehn function

We construct examples of finitely presented simple groups whose Dehn functions are at least exponential. To the best of our knowledge, these are the first such examples known. Our examples arise from Röver-Nekrashevych groups, using carefully calibrated self-similar representations of Baumslag-Solitar groups.

math.GR↗

arXiv2023

Finitely presented left orderable monsters

A left orderable monster is a finitely generated left orderable group all of whose fixpoint-free actions on the line are proximal: the action is semiconjugate to a minimal action so that for every bounded interval $I$ and open interval $J$, there is a group element that sends $I$ into $J$. In his 2018 ICM address, Navas asked about the existence of left orderable monsters. By now there are several examples, all of which are finitely generated but not finitely presentable. We provide the first examples of left orderable monsters that are finitely presentable, and even of type $F_\infty$. The construction itself is elementary, and these groups satisfy several additional properties separating them from the previous examples: they are not simple, they act minimally on the circle, and they have an infinite-dimensional space of homogeneous quasimorphisms. Our construction is flexible enough that it produces infinitely many isomorphism classes of finitely presented (and type $F_{\infty}$) left orderable monsters.

math.GR↗