Polynomially bounded cohomology and discrete groups

Abstract

We establish the homological foundations for studying polynomially bounded group cohomology, and show that the natural map from P H * (G; Q) to H * (G; Q) is an isomorphism for a certain class of groups.

Relatively hyperbolic groups, rapid decay algebras, and a generalization of the Bass conjecture

“…This result is shown in [Og2] for B D P , but the same argument works in this more general case (this point is also noted in [Me1] for P and E). …”

“…Let C z denote the infinite cyclic subgroup generated by z. Associated to the shortexact sequence of groups Z D C z G x G is a Serre spectral sequence in B-cohomology (the case B D P is done in detail in [Og2], the more general case follows by similar reasoning, with similar restrictions, cf. [Og3]).…”

“…Associated to the shortexact sequence of groups Z D C z G x G is a Serre spectral sequence in B-cohomology (the case B D P is done in detail in [Og2], the more general case follows by similar reasoning, with similar restrictions, cf. [Og3]). By a spectral sequence comparison together with induction on n, we see that nilpotent groups are B-isocohomological for all B Ã P .…”

“…If x is a non-elliptic conjugacy class, then the conjugacy bound in this case is linear [Gr], [Bu]. Moreover, G h is virtually cyclic and isometrically embedded in G for any h 2 x, implying that it is B-isocohomological [Ji1], [Og2]. It also implies that the quotient G h =.h/ is finite, so rcd.G h =.h// D 0.…”

“…Moreover, as we further show in this section, the cohomology theories identify with the corresponding B-bounded theory computed as the cohomology of the subcocomplex of B-bounded cochains in the cocomplex associated to the group algebra. This idea, which is a natural extension of both [Og2] and [M1], is most conveniently cast in the context of B-bounded cohomology theories associated to weighted simplicial sets (also defined in Section 1.3). To illustrate, there is an isomorphism…”