Fundamental groups and group presentations with bounded relator lengths

TL;DR

This paper investigates the geometric and spectral properties of spaces with trivial first Betti number under finite group actions, providing bounds on diameters, eigenvalues, and related invariants for Cayley graphs and CW-complexes.

Contribution

It establishes new bounds on diameters, eigenvalues, and Cheeger constants for spaces with trivial first Betti number under finite group actions, linking geometry and spectral graph theory.

Findings

Diameter ratio bound: diam(X)/diam(X/G) ≤ 4√|G|

Eigenvalue bound: λ₁ ≥ 2 - 2 cos(2π/k) for certain Cayley graphs

Explicit diameter upper bound: O(k²|S| log|G|)

Abstract

We study the geometry of compact geodesic spaces with trivial first Betti number admitting large finite groups of isometries. We show that if a finite group $G$ acts by isometries on a compact geodesic space $X$ whose first Betti number vanishes, then diam$(X) / $diam$(X / G ) \leq 4 \sqrt{ \vert G \vert }$. For a group $G$ and a finite symmetric generating set $S$, $P_k(\Gamma (G, S))$ denotes the 2-dimensional CW-complex whose 1-skeleton is the Cayley graph $\Gamma$ of $G$ with respect to $S$ and whose 2-cells are $m$-gons for $0 \leq m \leq k$, defined by the simple graph loops of length $m$ in $\Gamma$, up to cyclic permutations. Let $G$ be a finite abelian group with $\vert G \vert \geq 3$ and $S$ a symmetric set of generators for which $P_k(\Gamma (G,S))$ has trivial first Betti number. We show that the first nontrivial eigenvalue $-\lambda_1$ of the Laplacian on the Cayley graph satisfies $\lambda_1 \geq 2 - 2 \cos ( 2 \pi / k ) $. We also give an explicit upper bound on the diameter of the Cayley graph of $G$ with respect to $S$ of the form $O (k^2 \vert S \vert \log \vert G \vert )$. Related explicit bounds for the Cheeger constant and Kazhdan constant of the pair $(G,S)$ are also obtained.