Volume Entropy Rigidity for Random Groups at Low Densities
Dongming Hua
TL;DR
The work addresses volume-entropy rigidity for weighted word metrics on hyperbolic groups, extending Cantrell–Tanaka convexity to obtain existence and (up to rough isometry) uniqueness of a minimizing normalized weight. It introduces $\λ$-translation-apparent presentations to secure strict convexity and actual uniqueness, and shows that at low random-group densities these presentations are generic, with the unique minimizer near the uniform weight and the minimum entropy close to that of the free group. The analysis combines small-cancellation theory, Myers’ weighted subword counting, and probabilistic tools (Chernoff bounds) to establish both existence, uniqueness, and genericity results, along with a stability principle. The results illuminate how random groups at low density approximate free-group entropy under optimal weightings, with potential implications for understanding geometric and spectral rigidity in varied group-theoretic contexts.
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.