On the Howe--Moore property for automorphism groups of buildings

2026/06/26 by Pierre-Emmanuel Caprace, Andreas Thom
#math.GR #math.MG #math.OA

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Abstract

Let G be a closed type-preserving subgroup of the automorphism group of a thick locally finite building X of finite rank, and assume that G acts Weyl-transitively. We prove that every unitary representation of G is mixing, unless its restriction to a parabolic subgroup of minimal non-spherical type is amenable in the sense of Bekka. It follows that every unitary representation of G that is weakly contained in the regular representation, is mixing. In case X is of minimal non-spherical type and its thickness satisfies some modest lower bound, we deduce that G has the Howe--Moore property provided its only compact quotient is trivial. We also obtain results on rigidity of invariant random subgroups for Kac--Moody lattices of compact hyperbolic type, yielding examples of infinite finitely presented Kazhdan groups with exactly two ergodic invariant random subgroups.

Related

• Strong transitivity, Moufang's condition and the Howe--Moore property
• Scale and tidy subgroups for Weyl-transitive automorphism groups of buildings
• Rank one isometries of buildings and quasi-morphisms of Kac-Moody groups
• Open subgroups of the automorphism group of a right-angled building
• Rigidity of the group topology for closed Weyl transitive groups of automorphisms of a regular locally finite building