Critical exponents of invariant random subgroups in negative curvature

Abstract

Let X be a proper geodesic Gromov hyperbolic metric space and let G be a cocompact group of isometries of X admitting a uniform lattice. Let d be the Hausdorff dimension of the Gromov boundary ∂X. We define the critical exponent δ(µ) of any discrete invariant random subgroup µ of the locally compact group G and show that δ(µ) > d 2 in general and that δ(µ) = d if µ is of divergence type. Whenever G is a rank-one simple Lie group with Kazhdan's property (T ) it follows that an ergodic invariant random subgroup of divergence type is a lattice. One of our main tools is a maximal ergodic theorem for actions of hyperbolic groups due to Bowen and Nevo.

Stationary random subgroups in negative curvature

“…Unlike normal subgroups of lattices, there is a priori no reason for ν-generic subgroups to contain an entire conjugacy class of any particular hyperbolic element. Instead, we show in [GL19] that ν-almost every subgroup contains a positive proportion of some hyperbolic "approximate conjugacy class". More precisely, assume that the group Is(X) admits some auxiliary discrete subgroup Γ acting cocompactly on the space X.…”

“…was the main result of [GL19]. This is in turn a generalization of a result of Matsuzaki, 3 A probability measure µ on a locally compact group G is called spread out is some convolution power µ * n is non singular with respect to the Haar measure on the group G.…”

“…Divergence of Poincare series. The following uses the core arguments of [GL19] adapted to work in the context of random walks. The idea is to relate the critical exponents of ν-almost every discrete subgroup with the quantity δ(µ) 2 .…”

“…We expect our results concerning critical exponents can be extended beyond isometry groups of hyperbolic spaces, for instance to rank-one CAT(0)-spaces, Cayley graphs of relatively hyperbolic groups, and products of rank one symmetric spaces. The probabilistic framework for these generalizations is laid out in this paper, one needs to modify few geometric techniques from [GL19].…”

Co-induction and invariant random subgroups

“…The study of invariant random subgroups on various classes of groups has been an active area of research in the last several years, see, for example, [BGK17] and the references contained therein, as well as [Bo14], [TT-D14], [LM15], [Ge15], [BGN15], [O15], [LM15], [GL16], [HT16], [EG16], [BDLW16], [G17], [BBT17], [BT17], [DM17], [HY17], [Ge18], [BT18], [TT-D18], [BLT18], [GeL18]. One usually concentrates on the study of ergodic (with respect to the conjugacy action) invariant random subgroups, which are the extreme points of the Choquet simplex IRS(Γ).…”