arXiv subjects

Kajal Das

Publications and source records attributed to Kajal Das.

8 recordsLinked to original sources

arXiv2025

Invariance of non-vanishing of first $l^p$-cohomology under $L^q$-Measured Equivalence

The first $l^p$-cohomology is an algebro-analytical object attached to a finitely generated discrete group and introduced by M. Gromov. It is well known that it is invariant under quasi-isometry. In this article, we prove that the non-vanishing of the first $l^p$-cohomology of a non-amenable group is invariant under $L^q$-Measured Equiavalence (an equivalence relation introduced by Gromov), where $q\geq p$. We also discuss many applications of this result. We prove that for hyperbolic (in the sense of Gromov) Coxeter groups with boundaries having Combinatorial Loewner Property, conformal dimension (of the canonical conformal gauge) of the Gromov boundary is invariant under $L^q$-Measured Equivalence for some large $q$. We prove that the finitely generated free groups and surface groups are not $L^1$-Measured Equivalent. We also give a lower bound of the critical exponent for the first $l^p$-cohomology of any lattice in $SO(n,1)$. Finally, we discuss $L^q$-Measured Equivalence between non-amenable 3-manifold groups corresponding to Thurston's three geometries $\mathbb{H}^3$, $\mathbb{H}^2\times\mathbb{R}$ and $\widetilde{SL_2(\mathbb{R})}$.

math.GR↗

arXiv2024

Positivstellensatz for C*-tensor categories

We explore semi-pre-C*-algebras in the context of rigid semisimple C*-tensor categories and using techniques from annular representations, we extend Ozawa's criterion for property (T) in groups to this context

math.QA↗

arXiv2023

From the coarse geometry of warped cones to the measured coupling of groups

In this article, we prove that if two warped cones corresponding to two finitely generated groups with free, isometric, measure-preserving, actions on two compact metric spaces with probability measures are level-wise quasi-isometric (with some extra natural assumptions), then the corresponding groups are uniformly measured equivalent (UME). It was earlier known from the works of de Laat-Vigolo and Sawicki that if two such warped cones are level-wise quasi-isometric, then their stable products are quasi-isometric. We strengthen this result and go further to prove UME of the groups. We also discuss many applications of our main result. We give countably infinite examples of groups and associated Warped cones such that the groups are mutually quasi-isometric, but the Warped cones are not mutually quasi-isometric in the sense of our main theorem. We also provide examples of two Warped cones (which are quasi-isometric to two different expander families) such that one of them does not quasi-isometrically embed into the other one in the sense of our main theorem.

math.GR↗

arXiv2023

On Ahn-Hendrey-Kim-Oum question for twin-width of graphs with 6 vertices

Twin-width is a recently introduced graph parameter for finite graphs. It is an open problem to determine whether there is an $n$-vertex graph having twin-width at least $n/2$ (due to J. Ahn, K. Hendrey, D. Kim and S. Oum). In an earlier paper, the author showed that such a graph with less than equal to 5 vertices does not exist. In this article, we show that such a graph with 6 vertices does not exist. More precisely, we prove that each graph with 6 vertices has twin-width less than equal to 2.

math.CO↗

arXiv2023

On super-rigidity of Gromov's random monster group

In this article, we show super-rigidity of Gromov's random monster group. We prove that any morphism $ϕ_α$ from Gromov's random monster group $Γ_α$ to the group $G$ has finite image for almost all $α$, where $G$ is any of the following types of groups: mapping class group $MCG(S_{g,b})$, braid group $B_n$, outer automorphism group of a free group $Out(F_N)$, automorphism group of a free group $Aut(F_N)$, hierarchically hyperbolic group, a-$L^p$-menable group or K-amenable group. We introduce another property called hereditary super-rigidity and prove that $Γ_α$ has hereditary super-rigidity with respect to an a-$L^p$-menable group or a K-amenable group. We also establish a stability theorem for the groups with respect to which $Γ_α$ has super-rigidity and hereditary super-rigidity.

math.GR↗

arXiv2022

Computation of twin-width of graphs

Twin-width is a recently introduced graph parameter. In this article, we compute twin-width of various finite graphs. In particular, we prove that the twin-widths of finite graphs with 4 and 5 vertices are less than equal to 1 and 2, respectively. We show that the constructions of dual graph and line graph do not preserve twin-width. Also, we give upper bounds for the twin-width of King's graph and Rook's graph.

math.CO↗

arXiv2016

From the geometry of box spaces to the geometry and measured couplings of groups

In this paper, we prove that if two `box spaces' of two residually finite groups are coarsely equivalent, then the two groups are `uniform measured equivalent' (UME). More generally, we prove that if there is a coarse embedding of one box space into another box space, then there exists a `uniform measured equivalent embedding' (UME-embedding) of the first group into the second one. This is a reinforcement of the easier fact that a coarse equivalence (resp.\ a coarse embedding) between the box spaces gives rise to a coarse equivalence (resp.\ a coarse embedding) between the groups. We deduce new invariants that distinguish box spaces up to coarse embedding and coarse equivalence. In particular, we obtain that the expanders coming from $SL_n(\mathbb{Z})$ can not be coarsely embedded inside the expanders of $SL_m(\mathbb{Z})$, where $n>m$ and $n,m\geq 3$. Moreover, we obtain a countable class of residually groups which are mutually coarse-equivalent but any of their box spaces are not coarse-equivalent.

math.GR↗

arXiv2014

Integrable measure equivalence and the central extension of surface groups

Let $Γ_g$ be a surface group of genus $g\geq 2$. It is known that the canonical central extension $\tildeΓ_g$ and the direct product $Γ_g\times \mathbb{Z}$ are quasi-isometric. It is also easy to see that they are measure equivalent. By contrast, in this paper, we prove that quasi-isometry and measure equivalence cannot be achieved "in a compatible way". More precisely, these two groups are not uniform (nor even integrable) measure equivalent. In particular, they cannot act continuously, properly and cocompactly by isometries on the same proper metric space, or equivalently they are not uniform lattices in a same locally compact group.

math.MG↗