We establish a correspondence between the invariant subsets of a non-degenerate symmetric set-theoretical solution of the quantum Yang-Baxter equation and the parabolic subgroups of its structure group, equipped with its canonical Garside structure. Moreover, we introduce the notion of a foldable solution, which extends the one of a decomposable solution. AMS Subject Classification: 16T25, 20F36.

The Whitehead problem is solved in the class of toral relatively hyperbolic groups G (i.e. torsion-free relatively hyperbolic groups with abelian parabolic subgroups): there is an algorithm which, given two finite tuples (u1, . . . , un) and (v1, . . . , vn) of elements of G, decides whether there is an automorphism of G taking ui to vi for all i.

A class of finite monoids M constructed from a groupG of Lie type is considered. We describe the irreducible complex representations and prove the complete reducibility of the representations of M . The sandwich matrix of M is decomposed into a product of matrices corresponding to maximal parabolic subgroups of G.

We deal with two-generator subgroups of PSL(2, C) with real traces of both generators and their commutator. We give discreteness criteria for these groups when at least one of the generators is parabolic. We also present a list of the corresponding orbifolds. Mathematics Subject Classification (2000): Primary: 30F40; Secondary: 20H10, 22E40, 57M60, 57S30.

We study automata groups generated by reset automata. Every lamplighter group Z/nZ wr Z can be generated by such an automaton, and in general these automata groups are similar in nature to lamplighters: they are amenable locally-finite-by-cyclic groups; under mild decidable conditions, the semigroups generated by such automata are free. Parabolic subgroups and fractal properties are considered.

Let G = GL(n, F ) where F is a local field of arbitrary characteristic, and let π1, π2 be representations induced from characters of two maximal parabolic subgroups P1, P2. We explicitly determine the space HomG (π1, π2) of intertwining operators and prove that it has dimension ≤ 1 in all cases.

We describe geometric realizations for various classes of admissible representations of reductive Lie groups. The representations occur on partially holomorphic cohomology spaces corresponding to partially holomorphic homogeneous vector bundles over real group orbits in complex flag manifolds. The representations in question include standard tempered and limits of standard tempered representati...

The exceptional group G2 has two maximal parabolic subgroups Plong, Pshort corresponding to the so-called long root and short root. In this paper, the second author introduces two zeta functions associated to (G2, Plong) and (G2, Pshort) respectively, and the first author proves that these zetas satisfy the Riemann Hypothesis.

For a semisimple Lie group G with parabolic subgroups Q ⊂ P ⊂ G, we associate to a parabolic geometry of type (G,P ) on a smooth manifold N the correspondence space CN , which is the total space of a fiber bundle over N with fiber a generalized flag manifold, and construct a canonical parabolic geometry of type (G,Q) on CN . Conversely, for a parabolic geometry of type (G,Q) on a smooth manifol...

We define the notion of a “peripheral splitting” of a group. This is essentially a representation of the group as the fundamental group of a bipartite graph of groups, where all the vertex groups of one colour are held fixed—the “peripheral subgroups”. We develop the theory of such splittings and prove an accessibility result. The theory mainly applies to relatively hyperbolic groups with conne...