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Olga Varghese

Publications and source records attributed to Olga Varghese.

At least 19 recordsLinked to original sources
arXiv2026

Profinite properties of Coxeter groups

We prove a number of results about profinite completions of Coxeter groups. For example we prove Coxeter groups are good in the sense of Serre and that various splittings of Coxeter groups arising from actions on trees are detected by the profinite completion. As an application we prove a number of families of Coxeter groups are profinitely rigid amongst Coxeter groups. We also prove that Gromov-hyperbolic FC type, extra large type, and odd Coxeter groups are almost profinitely rigid amongst Coxeter groups. In the appendix, Sam Fisher and Sam Hughes show that the Atiyah Conjecture holds for all Coxeter groups, and that $\ell^2$-Betti numbers and their positive characteristic analogues are profinite invariants of Coxeter groups and of virtually compact special groups.

math.GR↗
arXiv2025

On the derived length of Dyer groups

By definition, a group $G$ is quasi-perfect, if $G$ is perfect or the commutator subgroup of $G$ is perfect. In this note we give a description of quasi-perfect Dyer groups by properties of the corresponding Dyer graphs.

math.GR↗
arXiv2025

Higman-Thompson groups and profinite properties of right-angled Coxeter groups

We prove that every right-angled Coxeter group (RACG) is profinitely rigid amongst all Coxeter groups. On the other hand we exhibit RACGs which have infinite profinite genus amongst all finitely generated residually finite groups. We also establish profinite rigidity results for graph products of finite groups. Along the way we prove that the Higman-Thompson groups $V_{n}$ are generated by $4$ involutions, generalising a classical result of Higman for Thompson's group $V$.

math.GR↗
arXiv2024

Involutions in Coxeter groups

We combinatorially characterize the number $\mathrm{cc}_2$ of conjugacy classes of involutions in any Coxeter group in terms of higher rank odd graphs. This notion naturally generalizes the concept of odd graphs, used previously to count the number of conjugacy classes of reflections. We provide uniform bounds and discuss some extremal cases, where the number $\mathrm{cc}_2$ is smallest or largest possible. Moreover, we provide formulae for $\mathrm{cc}_2$ in free and direct products as well as for some finite and affine types, besides computing $\mathrm{cc}_2$ for all triangle groups, and all affine irreducible Coxeter groups of rank up to eleven.

math.GR↗
arXiv2023

The growth series of Dyer groups

Graph products of cyclic groups and Coxeter groups are two families of groups that are defined by labeled graphs. The family of Dyer groups contains these both families and gives us a framework to study these groups in a unified way. This paper focuses on the growth series of a Dyer group $D$ with respect to the standard generating set. We give a recursive formula for the growth series of $D$ in terms of the growth series of standard parabolic subgroups. As an application we obtain the rationality of the growth series of a Dyer group. Furthermore, we show that the growth series of $D$ is closely related to the Euler characteristic of $D$.

math.GR↗
arXiv2023

On quotients of Coxeter groups

A group $G$ is said to be just infinite if $G$ itself is infinite but all proper quotients of $G$ are finite. We show that a Coxeter group $W_Γ$ is just infinite if and only if $Γ$ is isomorphic to one of the following graphs: $\widetilde{A}_1$, $\widetilde{A}_n (n\geq 2)$, $\widetilde{B}_n (n\geq 3)$, $\widetilde{C}_n (n\geq 2)$, $\widetilde{D}_n(n\geq 4)$, $\widetilde{E}_6$, $\widetilde{E}_7$, $\widetilde{E}_8$, $\widetilde{F}_4$ or $\widetilde{G}_2$. Moreover, we show that just infinite Coxeter groups are profinitely rigid among all Coxeter groups.

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arXiv2023

Steep uncountable groups

We produce a simple group $G$ of cardinality $\aleph_1$ which is Artinian (every strictly descending chain of subgroups is finite), satisfies a Burnside law and such that for each uncountable subset $Y \subseteq G$ there exists a natural number $n_Y$ for which every element of $G$ may be expressed as a product of length at most $n_Y$ of elements in $Y^{\pm 1}$. In particular this group is Jónsson (every proper subgroup is of strictly smaller cardinality) and strongly bounded (every abstract action on a metric space has bounded orbits); this is the first example of an uncountable group having both of these properties which is constructed without using the continuum hypothesis. The group $G$ can also be made so that all subgroups are simple and all nontrivial subgroups are malnormal in $G$.

math.GR↗
arXiv2023

On normal subgroups in automorphism groups

We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group ${\rm Aut}(A_Γ)$. In particular, we prove that a finite normal subgroup in ${\rm Aut}(A_Γ)$ has at most order two and if $Γ$ is not a clique, then any finite normal subgroup in ${\rm Aut}(A_Γ)$ is trivial. This property has implications to automatic continuity and to $C^\ast$-algebras: every algebraic epimorphism $φ\colon L\twoheadrightarrow{\rm Aut}(A_Γ)$ from a locally compact Hausdorff group $L$ is continuous if and only if $A_Γ$ is not isomorphic to $\mathbb{Z}^n$ for any $n\geq 1$. Further, if $Γ$ is not a join and contains at least two vertices, then the set of invertible elements is dense in the reduced group $C^\ast$-algebra of Aut$(A_Γ)$. We obtain similar results for ${\rm Aut}(G_Γ)$ where $G_Γ$ is a graph product of cyclic groups. Moreover, we give a description of the center of Aut$(G_Γ)$ in terms of the defining graph $Γ$.

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arXiv2022

Coxeter quotients of the automorphism group of a Coxeter group

We show that for a large class $\mathcal{W}$ of Coxeter groups the following holds: Given a group $W_Γ$ in $\mathcal{W}$, the automorphism group ${\rm Aut}(W_Γ)$ virtually surjects onto some infinite Coxeter group. In particular, the group ${\rm Aut}(W_Γ)$ is virtually indicable and therefore does not have Kazhdan's property (T).

math.GR↗
arXiv2022

A note on semicompleteness of graph products of abelian groups

In this short note we prove that a graph product $G_Γ$ of finitely generated abelian groups is semicomplete -- that is the kernel of the natural homomorphism ${\rm Aut}(G_Γ)\to{\rm Aut}(G_Γ^{ab})$ induced by the abelianization of $G_Γ$ is equal to the inner automorphisms -- if and only if $Γ$ does not have a separating star.

math.GR↗
arXiv2022

Automatic continuity for groups whose torsion subgroups are small

We prove that a group homomorphism $φ\colon L\to G$ from a locally compact Hausdorff group $L$ into a discrete group $G$ either is continuous, or there exists a normal open subgroup $N\subseteq L$ such that $φ(N)$ is a torsion group provided that $G$ does not include $\mathbb{Q}$ or the $p$-adic integers $\mathbb{Z}_p$ or the Prüfer $p$-group $\mathbb{Z}(p^\infty)$ for any prime $p$ as a subgroup, and if the torsion subgroups of $G$ are small in the sense that any torsion subgroup of $G$ is artinian. In particular, if $φ$ is surjective and $G$ additionaly does not have non-trivial normal torsion subgroups, then $φ$ is continuous. As an application we obtain results concerning the continuity of group homomorphisms from locally compact Hausdorff groups to many groups from geometric group theory, in particular to automorphism groups of right-angled Artin groups and to Helly groups.

math.GR↗
arXiv2022

On parabolic subgroups of Artin groups

Given an Artin group $A_Γ$, a common strategy in the study of $A_Γ$ is the reduction to parabolic subgroups whose defining graphs have small diameter, i.e. showing that $A_Γ$ has a specific property if and only if all "small" parabolic subgroups of $A_Γ$ have this property. Since "small" parabolic subgroups are the puzzle pieces of $A_Γ$ one needs to study their behavior, in particular their intersections. The conjecture we address here says that the class of parabolic subgroups of $A_Γ$ is closed under intersection. Under the assumption that intersections of parabolic subgroups in complete Artin groups are parabolic, we show that the intersection of a complete parabolic subgroup with an arbitrary parabolic subgroup is parabolic. Further, we connect the intersection behavior of complete parabolic subgroups of $A_Γ$ to fixed point properties and to automatic continuity of $A_Γ$ using Bass-Serre theory and a generalization of the Deligne complex.

math.GR↗
arXiv2021

Abstract group actions of locally compact groups on CAT(0) spaces

We study abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion free CAT(0) group is continuous.

math.GR↗
arXiv2020

On number of ends of graph products of groups

Given a finite simplicial graph $Γ=(V,E)$ with a vertex-labelling $φ:V\rightarrow\left\{\text{non-trivial finitely generated groups}\right\}$, the graph product $G_Γ$ is the free product of the vertex groups $φ(v)$ with added relations that imply elements of adjacent vertex groups commute. For a quasi-isometric invariant $\mathcal{P}$, we are interested in understanding under which combinatorial conditions on the graph $Γ$ the graph product $G_Γ$ has property $\mathcal{P}$. In this article our emphasis is on number of ends of a graph product $G_Γ$. In particular, we obtain a complete characterization of number of ends of a graph product of finitely generated groups.

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arXiv2019

A Nunke type classification in the locally compact setting

In this short note we prove that a group G is lcH-slender -- that is, every abstract group homomorphism from a locally compact Hausdorff topological group to G has an open kernel -- if and only if G is torsion-free and does not include Q or the p-adic integers Zp for any prime p. This mirrors a classical characterization given by Nunke for slender abelian groups.

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arXiv2019

The automorphism group of the universal Coxeter group

We study fixed point properties of the automorphism group of the universal Coxeter group Aut$(W_n)$. In particular, we prove that whenever Aut$(W_n)$ acts by isometries on complete $d$-dimensional CAT$(0)$ space with $d<\lfloor\frac{n}{2}\rfloor$, then it must fix a point. We also prove that Aut$(W_n)$ does not have Kazhdan's property (T). Further, strong restrictions are obtained on homomorphisms of Aut$(W_n)$ to groups that do not contain a copy of Sym(n).

math.GR↗