Volume Entropy Rigidity for Random Groups at Low Densities

TL;DR

This paper investigates the rigidity of volume entropy in hyperbolic groups with weighted metrics, demonstrating that at low densities, random groups typically have a unique entropy-minimizing weight close to uniform, revealing a generic structural property.

Contribution

It establishes conditions for unique entropy minimizers in hyperbolic groups and shows these conditions are generically satisfied in random groups at low densities, connecting entropy rigidity with probabilistic group theory.

Findings

Unique normalized weight minimizing entropy exists under certain conditions.

Conditions for entropy rigidity are generic for random groups at small densities.

The entropy minimizer for such groups is close to the uniform weight.

Abstract

We study the rigidity of the volume entropy for weighted word metrics on hyperbolic groups, building on a recent convexity result due to Cantrell-Tanaka. Using ideas from small cancellation theory, we give conditions under which a hyperbolic group admits a unique normalized weight minimizing the entropy. Moreover, we show that these conditions are generic for random groups at small densities, and that the unique minimizer of such a generic group is arbitrarily close to the uniform weight.