On Radon measures invariant under horospherical flows on geometrically infinite quotients

TL;DR

This paper investigates Radon measures invariant under horospherical flows on geometrically infinite quotients of hyperbolic space, showing such measures are supported on points with degenerate trajectories unless additional invariance is present.

Contribution

It characterizes invariant Radon measures on infinite hyperbolic quotients, revealing their support on degenerate trajectories unless extra invariance properties are assumed.

Findings

Invariant measures are supported on degenerate trajectories.

Measures without extra invariance lack support on regular trajectories.

The results extend understanding of measure invariance in infinite hyperbolic quotients.

Abstract

We consider a locally finite (Radon) measure on $ SO^+(d,1)/ \Gamma $ invariant under a horospherical subgroup of $ SO^+(d,1) $ where $ \Gamma $ is a discrete, but not necessarily geometrically finite, subgroup. We show that whenever the measure does not observe any additional invariance properties then it must be supported on a set of points with geometrically degenerate trajectories under the corresponding contracting $ 1 $-parameter diagonalizable flow (geodesic flow).