On centralizers of generalized uniform subgroups of locally compact groups

Abstract

Let G G be a locally compact group and H H a closed subgroup of G G such that the homogeneous space G / H G/H admits a finite invariant measure. Let Z G ( H ) {Z_G}(H) be the centralizer of H H in G G . It is shown that if G G is connected then Z G ( H ) {Z_G}(H) modulo its center is compact. If G G is only assumed to be locally connected it is shown that the commutator subgroup of Z G ( H ) {Z_G}(H) has compact closure. Consequences of these results are found for special classes of groups, such as Lie groups. An example of a totally disconnected group G G is given to show that the results for Z G ( H ) {Z_G}(H) need not hold if G G is not connected or locally connected.