arXiv subjects

Aditya De Saha

Publications and source records attributed to Aditya De Saha.

4 recordsLinked to original sources

arXiv2026

On the Coarse Lusternik-Schnirelmann Category of Groups

We introduce a coarse analog of the classical Lusternik-Schnirelmann category which we denote by $\text{c-cat}$, defined for metric spaces in the coarse homotopy category. This provides a new tool for studying large-scale topological properties of groups and spaces. We establish that $\text{c-cat}$ is a coarse homotopy invariant and prove a lower-bound $\text{p-cat}(Γ)\leq \text{c-cat}(Γ)$ for geometrically finite groups $Γ$, where $\text{p-cat}$ denotes the proper LS-category introduced in 1992 by Ayala and co-authors. We also prove an upper bound $\text{c-cat}(Γ) \leq \text{asdim}(Γ)$ for bicombable 1-ended groups which are semistable at $\infty$.

math.GT↗

arXiv2025

Fixed-point-free involutions on the boundaries of Right-angled Coxeter Groups

About 20 years ago, Bogdan Nica conjectured that the boundary of any word-hyperbolic group admits admits a fixed-point-free involution. In this very short article, we prove a variation of the conjecture, replacing word-hyperbolic groups with Right-angled coxeter groups. This is not a solution to Nica's conjecture, but hopefully the techniques will shed some light regarding possible approaches to the problem.

math.GR↗

arXiv2024

Persistent equivariant cohomology

This article has two goals. First, we hope to give an accessible introduction to persistent equivariant cohomology. Given a topological group $G$ acting on a filtered space, persistent Borel equivariant cohomology measures not only the shape of the filtration, but also attributes of the group action on the filtration, including in particular its fixed points. Second, we give an explicit description of the persistent equivariant cohomology of the circle action on the Vietoris-Rips metric thickenings of the circle, using the Serre spectral sequence and the Gysin homomorphism. Indeed, if $\frac{2πk}{2k+1} \le r < \frac{2π(k+1)}{2k+3}$, then $H^*_{S^1}(\mathrm{VR}^\mathrm{m}(S^1;r))\cong \mathbb{Z}[u]/(1\cdot3\cdot5\cdot\ldots \cdot (2k+1)\, u^{k+1})$ where $\mathrm{deg}(u)=2$.

math.AT↗

arXiv2023

On cohomological dimension of group homomorphisms

The (co)homological dimension of homomorphism $ϕ:G\to H$ is the maximal number $k$ such that the induced homomorphism is nonzero for some $H$-module. The following theorems are proven: THEOREM 1. For every homomorphism $ϕ:G\to H$ of a geometrically finite group $G$ the homological dimension of $ϕ$ equals the cohomological dimension, $hd(ϕ)= cd(ϕ)$. THEOREM 2. For every homomorphism $ϕ:G\to H$ of geometrically finite groups $cd(ϕ\timesϕ)=2cd(ϕ)$.

math.AT↗