In his paper, ’On torsion in the cohomology of locally symmetric varieties’, Peter Scholze has introduced a new, purely topological method to construct classes on arithmetic quotients spaces reductive groups over $$\mathbb {Q}$$ originating from similar Levi subgroups maximal parabolic subgroups. We extend this construction beyond cases he considers, and, complex case, local systems.

Given a class of compact spaces, we ask which groups can be maximal parabolic subgroups of a relatively hyperbolic group whose boundary is in the class. We investigate the class of 1-dimensional connected boundaries. We get that any non-torsion infinite f.g. group is a maximal parabolic subgroup of some relatively hyperbolic group with connected one-dimensional boundary without global cut point...

If (π, V ) is an admissible representation of a p-adic reductive group G and P = MN is a parabolic subgroupwith unipotent radicalN , its Jacquet module VN is the universalN -trivial quotientH0(N,V ) of V by the span of vectors π(n)v − v (n ∈ N ). One fundamental property is that V VN is an exact functor. This construction plays an important role in homomorphisms into representations induced fro...

In this paper, we investigate boundaries of parabolic subgroups of Coxeter groups. Let (W, S) be a Coxeter system and let T be a subset of S such that the parabolic subgroup WT is infinite. Then we show that if a certain set is quasi-dense in W , then W∂Σ(WT , T ) is dense in the boundary ∂Σ(W, S) of the Coxeter system (W, S), where ∂Σ(WT , T ) is the boundary of (WT , T ).

3.3. The greatest common divisor and the least common multiple 12 3.4. Parabolic divisors 14 3.5. Divisibility: further properties 14 4. Normal forms arising from HNN extensions 16 4.1. HNN -normal form 16 5. Conjugacy problem 18 5.1. Conjugacy Criterion for HNN -extensions 18 5.2. Block decomposition 22 5.3. Conjugacy with respect to a subgroup 23 5.4. An algorithm for solving the Conjugacy Pr...