Combable groups have group cohomology of polynomial growth

Group cohomology of polynomial growth is defined for any finitely generated discrete group, using cochains that have polynomial growth with respect to the word length function. We give a geometric condition that guarantees that it agrees with the usual group cohomology and verify this condition for a class of combable groups. Our condition involves a chain complex that is closely related to exotic cohomology theories studied by Allcock and Gersten and by Mineyev.Comment: 19 pages, typo corrected in version

Slant products on the Higson-Roe exact sequence

We construct a slant product $/ \colon \mathrm{S}_p(X \times Y) \times \mathrm{K}_{1-q}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{S}_{p-q}(X)$ on the analytic structure group of Higson and Roe and the K-theory of the stable Higson corona of Emerson and Meyer. The latter is the domain of the co-assembly map $\mu^\ast \colon \mathrm{K}_{1-\ast}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{K}^\ast(Y)$. We obtain such products on the entire Higson--Roe sequence. They imply injectivity results for external product maps. Our results apply to products with aspherical manifolds whose fundamental groups admit coarse embeddings into Hilbert space. To conceptualize the class of manifolds where this method applies, we say that a complete $\mathrm{spin}^{\mathrm{c}}$-manifold is Higson-essential if its fundamental class is detected by the co-assembly map. We prove that coarsely hypereuclidean manifolds are Higson-essential. We draw conclusions for positive scalar curvature metrics on product spaces, particularly on non-compact manifolds. We also obtain equivariant versions of our constructions and discuss related problems of exactness and amenability of the stable Higson corona.Comment: 82 pages; v2: Minor improvements. To appear in Ann. Inst. Fourie

GEOMETRY OF THE WORD PROBLEM FOR 3-MANIFOLD GROUPS

We provide an algorithm to solve the word problem in all fundamental groups of 3-manifolds that are either closed, or compact with (finitely many) boundary compo- nents consisting of incompressible tori, by showing that these groups are autostackable. In particular, this gives a common framework to solve the word problem in these 3-manifold groups using finite state automata. We also introduce the notion of a group which is autostackable respecting a subgroup, and show that a fundamental group of a graph of groups whose vertex groups are autostackable respecting any edge group is autostackable. A group that is strongly coset automatic over an autostackable subgroup, using a prefix-closed transversal, is also shown to be autostackable respecting that subgroup. Building on work by Antolin and Ciobanu, we show that a finitely generated group that is hyperbolic relative to a collection of abelian subgroups is also strongly coset automatic relative to each subgroup in the collection. Finally, we show that fundamental groups of compact geometric 3-manifolds, with boundary consisting of (finitely many) incompressible torus components, are autostackable respecting any choice of peripheral subgroup