arXiv subjects

Olga Varghese

Publications and source records attributed to Olga Varghese.

30 records · Page 2Linked to original sources

arXiv2019

Abstract homomorphisms from locally compact groups to discrete groups

We show that every abstract homomorphism $φ$ from a locally compact group $L$ to a graph product $G_Γ$, endowed with the discrete topology, is either continuous or $φ(L)$ lies in a 'small' parabolic subgroup. In particular, every locally compact group topology on a graph product whose graph is not 'small' is discrete. This extends earlier work by Morris-Nickolas. We also show the following. If $L$ is a locally compact group and if $G$ is a discrete group which contains no infinite torsion group and no infinitely generated abelian group, then every abstract homomorphism $φ:L\to G$ is either continuous, or $φ(L)$ is contained in the normalizer of a finite nontrivial subgroup of $G$. As an application we obtain results concerning the continuity of homomorphisms from locally compact groups to Artin and Coxeter groups.

math.GR↗

arXiv2019

Conjugating automorphisms of graph products: Kazhdan's property (T) and SQ-universality

An automorphism of a graph product of groups is conjugating if it sends each factor to a conjugate of a factor (possibly different). In this article, we determine precisely when the group of conjugating automorphisms of a graph product satisfies Kazhdan's property (T) and when it satisfies some vastness properties including SQ-universality.

math.GR↗

arXiv2019

On hyperbolicity and virtual freeness of automorphism groups

We show that word hyperbolicity of automorphism groups of graph products $G_Γ$ and of Coxeter groups $W_Γ$ depends strongly on the shape of the defining graph $Γ$. We also characterized those $Aut(G_Γ)$ and $Aut(W_Γ)$ in terms of $Γ$ that are virtually free.

math.GR↗

arXiv2019

Planarity of Cayley graphs of graph products of groups

We obtain a complete classification of graph products of finite abelian groups whose Cayley graphs with respect to the standard presentations are planar.

math.GR↗

arXiv2019

On property (T) for automorphism groups of graph products

We show that the automorphism group of a graph product of finite groups $Aut(G_Γ)$ has Kazhdan's property (T) if and only if $Γ$ is a complete graph.

math.GR↗

arXiv2018

A note on locally elliptic actions on cube complexes

We deduce from Sageev's results that whenever a group acts locally elliptically on a finite dimensional CAT(0) cube complex, then it must fix a point. As an application, we give an example of a group G such that G does not have property (T), but G and all its finitely generated subgroups can not act without a fixed point on a finite dimensional CAT(0) cube complex, answering a question by Barnhill and Chatterji.

math.GR↗

arXiv2018

On coherence of graph products and Coxeter groups

We study coherence of graph products and Coxeter groups and obtain many results in this direction.

math.GR↗

arXiv2018

A condition that prevents groups from acting fixed point free on cube complexes

We describe a group theoretic condition which ensures that any cellular action of a group satisfying this condition on a CAT(0) cube complex has a global fixed point. In particular, we show that this fixed point criterion is satisfied by the automorphism group of a free group of rank n. For the unique subgroup of index two in the automorphism group of a free group, we obtain a similar result.

math.GR↗

arXiv2018

Representations of groups with CAT(0) fixed point property

We show that certain representations over fields with positive characteristic of groups having CAT(0) fixed point property ${\rm F}\mathcal{B}_{\widetilde{A}_n}$ have finite image. In particular, we obtain rigidity results for representations of the following groups: the special linear group over $\mathbb{Z}$, $SL_k(\mathbb{Z})$, the special automorphism group of a free group, ${\rm SAut}(F_k)$, the mapping class group of a closed orientable surface, ${\rm Mod}(Σ_g)$, and many other groups. In the case of characteristic zero we show that low dimensional complex representations of groups having CAT(0) fixed point property ${\rm F}\mathcal{B}_{\widetilde{A}_n}$ have finite image if they always have compact closure.

math.GR↗

arXiv2015

SAut(Fn) cannot act on small tori

We study smooth actions of SAut(Fn), the unique subgroup of index two in the automorphism group of a free group of rank n, as a part of the generalized 'Zimmer program'. In particular, we show that every action of SAut(Fn) on a low dimensional torus is trivial.

math.GR↗

arXiv2015

Low dimensional linear representations of SAut(Fn)

We prove that SAut(Fn), the unique subgroup of index two in the automorphism group of a free group of rank n, admits no non-trivial linear representation of degree d<n for any field of characteristic not equal to two.

math.GR↗

arXiv2015

Fixed points for actions of Aut(Fn) on CAT(0) spaces

For n greater or equal 4 we discuss questions concerning global fixed points for isometric actions of Aut(Fn), the automorphism group of a free group of rank n, on complete CAT(0) spaces. We prove that whenever Aut(Fn) acts by isometries on complete d-dimensional CAT(0) space with d is less than 2 times the integer function of n over 4 and minus 1, then it must fix a point. This property has implications for irreducible representations of Aut(Fn), which are also presented here. For SAut(Fn), the unique subgroup of index two in Aut(Fn), we obtain similar results.

math.GR↗