Hypercontractivity in group von Neumann algebras

TL;DR

This paper introduces combinatorial and numerical methods to establish hypercontractivity estimates in group von Neumann algebras, with applications to various groups including free, triangular, cyclic, Coxeter, and groups with property (T).

Contribution

It presents new combinatorial and numerical techniques for hypercontractivity in group von Neumann algebras, including optimal bounds for specific groups and broader inequalities for classes of groups.

Findings

Optimal hypercontractive inequalities for free, triangular, and cyclic groups.

A general method for $L_p o L_q$ hypercontractivity via logarithmic Sobolev inequalities.

Hypercontractivity fails for groups with Kazhdan property (T).

Abstract

In this paper, we provide a combinatorial/numerical method to establish new hypercontractivity estimates in group von Neumann algebras. We will illustrate our method with free groups, triangular groups and finite cyclic groups, for which we shall obtain optimal time hypercontractive $L_2 \to L_q$ inequalities with respect to the Markov process given by the word length and with $q$ an even integer. Interpolation and differentiation also yield general $L_p \to L_q$ hypercontrativity for $1 < p \le q < \infty$ via logarithmic Sobolev inequalities. Our method admits further applications to other discrete groups without small loops as far as the numerical part ---which varies from one group to another--- is implemented and tested in a computer. We also develop another combinatorial method which does not rely on computational estimates and provides (non-optimal) $L_p \to L_q$ hypercontractive inequalities for a larger class of groups/lengths, including any finitely generated group equipped with a conditionally negative word length, like infinite Coxeter groups. Our second method also yields hypercontractivity bounds for groups admitting a finite dimensional proper cocycle. Hypercontractivity fails for conditionally negative lengths in groups satisfying Kazhdan property (T).