Nimish A. Shah

Limit distributions of expanding translates of certain orbits on homogeneous spaces

AbstractLetL be a Lie group and λ a lattice inL. SupposeG is a non-compact simple Lie group realized as a Lie subgroup ofL and

EQUIDISTRIBUTION AND COUNTING FOR ORBITS OF GEOMETRICALLY FINITE HYPERBOLIC GROUPS

\overline {GA} = L

Equidistribution of expanding translates of curves and Dirichlet’s theorem on diophantine approximation

. LetaεG be such that Ada is semisimple and not contained in a compact subgroup of Aut(Lie(G)). Consider the expanding horospherical subgroup ofG associated toa defined as U+ ={gεG:a−n gan} →e asn → ∞. Let Ω be a non-empty open subset ofU+ andni → ∞ be any sequence. It is showed that

Strong wavefront lemma and counting lattice points in sectors

\overline { \cup _{i = 1}^\infty a^n \Omega \Lambda } = L

LIMITING DISTRIBUTIONS OF CURVES UNDER GEODESIC FLOW ON HYPERBOLIC MANIFOLDS

. A stronger measure theoretic formulation of this result is also obtained. Among other applications of the above result, we describeG-equivariant topological factors of L/gl × G/P, where the real rank ofG is greater than 1,P is a parabolic subgroup ofG andG acts diagonally. We also describe equivariant topological factors of unipotent flows on finite volume homogeneous spaces of Lie groups.

Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds

Let G be a Lie group and Γ be a discrete subgroup. We show that if {μ n } is a convergent sequence of probability measures on G/Γ which are invariant and ergodic under actions of unipotent one-parameter subgroups, then the limit μ of such a sequence is supported on a closed orbit of the subgroup preserving it, and is invariant and ergodic for the action of a unipotent one-parameter subgroup of G .