Unitary L p+ -representations of almost automorphism groups

Let G be a locally compact group with an open subgroup H with the Kunze–Stein property, and let π be a unitary representation of H. We show that the representation π ˜ of G induced from π is an L p+ -representation if and only if π is an L p+ -representation. We deduce the following consequence for a large natural class of almost automorphism groups G of trees: For every p∈(2,∞), the group G has a unitary L p+ -representation that is not an L q+ -representation for any q<p. This in particular applies to the Neretin groups.