Groups, Geometry, and Dynamics

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Volume 9, Issue 1, 2015, pp. 29–53
DOI: 10.4171/GGD/304

Published online: 2015-04-28

On the growth of Hermitian groups

Rui Palma[1]

(1) University of Oslo, Norway

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.

Keywords: Hermitian group, growth rate of groups

Palma Rui: On the growth of Hermitian groups. Groups Geom. Dyn. 9 (2015), 29-53. doi: 10.4171/GGD/304