Involutions on Kleinian groups

Computing arithmetic Kleinian groups

Arithmetic Kleinian groups are arithmetic lattices in PSL2(C). We present an algorithm that, given such a group Γ, returns a fundamental domain and a finite presentation for Γ with a computable isomorphism.

Complex projective structures on Kleinian groups

Let M be a compact, oriented, irreducible, and boundary incompressible 3–manifold. Assume that its fundamental group is without rank two abelian subgroups and ∂M 6= ∅. We will show that every homomorphism θ: π1(M )→ PSL(2,C) which is not “boundary elementary” is induced by a possibly branched complex projective structure on the boundary of a hyperbolic manifold homeomorphic to M . AMS Classific...

Classification of Kleinian Groups

We present here a complete classification of those Kleinian groups which have an invariant region of discontinuity and which, in their action on hyperbolic 3-space, have a finite-sided fundamental polyhedron. This classification is complete in the same sense that finitely-generated Fuchsian groups of the first kind are completely classified, i.e., there is a countable collection of topologicall...

Commensurability riteria for Kleinian groups

Involutions in Weyl Groups

Let G be a split real group with Weyl group W . Let E be an irreducible representation of W . Let V be the stable Lie algebra version of the coherent continuation representation of W . The main result of this paper is a formula for the multiplicity of E in V . The formula involves the position of E in Lusztig’s set ∐ M(G). The paper treats all quasi-split groups G as well.