Measures Invariant under Horospherical Subgroups in Positive Characteristic

On Discrete Subgroups Containing a Lattice in a Horospherical Subgroup

We showed in [Oh] that for a simple real Lie group G with real rank at least 2, if a discrete subgroup Γ of G contains lattices in two opposite horospherical subgroups, then Γ must be a non-uniform arithmetic lattice in G, under some additional assumptions on the horospherical subgroups. Somewhat surprisingly, a similar result is true even if we only assume that Γ contains a lattice in one horo...

A Joining Classification and a Special Case of Raghunathan’s Conjecture in Positive Characteristic (with an Appendix by Kevin Wortman)

We prove the classification of joinings for maximal horospherical subgroups acting on homogeneous spaces without any restriction on the characteristic. Using the linearization technique we deduce a special case of Raghunathan’s orbit closure conjecture. In the appendix quasi-isometries of higher rank lattices in semisimple algebraic groups over fields of positive characteristic are characterized.

Harmonic and Invariant Measures on Foliated Spaces

We consider the family of harmonic measures on a lamination L of a compact space X by locally symmetric spaces L of noncompact type, i.e. L ∼= ΓL\G/K. We establish a natural bijection between these measures and the measures on the associated lamination by G-orbits, Ò L , which are right invariant under a minimal parabolic (Borel) subgroup B < G. In the special case when G is split, these measur...

S-arithmeticity of Discrete Subgroups Containing Lattices in Horospherical Subgroups

0. Introduction. Let Qp be the field of p-adic numbers, and let Q∞ = R. Let Gp be a connected semisimpleQp-algebraic group. The unipotent radical of a proper parabolic Qp-subgroup of Gp is called a horospherical subgroup. Two horospherical subgroups are called opposite if they are the unipotent radicals of two opposite parabolic subgroups. In [5] and [6], we studied discrete subgroups generated...

On Quasi-Invariant Transverse Measures for the Horospherical Foliation of a Negatively Curved Manifold