Let G be a Garside group endowed with the generating set $${\cal S}$$ of non-trivial simple elements, and let H parabolic subgroup G. We determine transversal T in such that each θ ∈ is minimal length its right-coset, Hθ, for word respect to . show there exists regular language L on S} \cup {{\cal S}^{- 1}}$$ bijection ev: → satisfying $$\lg \left(U \right) = {\lg _{\cal S}}\left({ev\left(U \ri...

Let G be a simple algebraic group of classical type over an algebraically closed field k. Let P be a parabolic subgroup of G and let p = LieP be the Lie algebra of P with Levi decomposition p = l⊕ u, where u is the Lie algebra of the unipotent radical of P and l is a Levi complement. Thanks to a fundamental theorem of R. W. Richardson ([16]), P acts on u with an open dense orbit; this orbit is ...

Lattices and parabolic subgroups are the obvious examples of cocompact subgroups of a connected, semisimple Lie group with finite center. We use an argument of C. C. Moore to show that every cocompact subgroup is, roughly speaking, a combination of these. We study a cocompact subgroup H of a connected, semisimple Lie group G with finite center. The case where H is discrete is very important and...

Let G be a rank 2 complete affine or hyperbolic Kac–Moody group over a finite field k. Then G is locally compact and totally disconnected. Let B− = HU− be the negative minimal parabolic subgroup of G, where H is the analog of a diagonal subgroup and U− is generated by all negative real root groups. Let w1 and w2 be the generators of the Weyl group. Let P − 1 = B − t Bw1B be the negative standar...

We construct first examples of discrete geometrically finite subgroups of PU(2, 1) which contain parabolic elements, and are isomorphic to surface groups of genus ≥ 2.