On the growth of Hermitian groups

TL;DR

This paper investigates the growth constraints of Hermitian groups, establishing bounds that distinguish them from non-Hermitian groups and providing new examples of such groups with specific growth properties.

Contribution

It introduces a novel bound on the growth of Hermitian groups using Banach algebra capacity, expanding the classification of non-Hermitian groups.

Findings

Bound on the growth rate of Hermitian groups

Identification of new non-Hermitian groups including free Burnside groups and p-adic groups

Extension of existing theory to broader classes of groups

Abstract

A locally compact group G is said to be Hermitian if every selfadjoint element of L^1(G) has real spectrum. Using Halmos' notion of capacity in Banach algebras and a result of Jenkins, Fountain, Ramsay and Williamson we will put a bound on the growth of Hermitian groups. In other words, we will show that if G has a subset that grows faster than a certain constant, then G cannot be Hermitian. Our result allows us to give new examples of non-Hermitian groups which could not tackled by the existing theory. The examples include certain infinite free Burnside groups, automorphism groups of trees, and p-adic general and special linear groups.