For a finitely-presented, torsion-free, discrete group G, the Atiyah conjecture asserts that the L-Betti numbers of any finite CW-complex with fundamental group G are integers; this conjecture has a natural extension to all groups. We prove that the class of groups for which the (extended) Atiyah conjecture holds and the finite subgroups have only finitely many different orders, is closed under...

We discuss time complexity of The Conjugacy Problem in HNN-extensions of groups, in particular, in Miller’s groups. We show that for “almost all”, in some explicit sense, elements, the Conjugacy Problem is decidable in cubic time. It is worth noting that the Conjugacy Problem in a Miller group may have be undecidable. Our results show that “hard” instances of the problem comprise a negligibly s...

It is shown that the knapsack problem, which was introduced by Myasnikov et al. for arbitrary finitely generated groups, can be solved in NP for graph groups. This result even holds if the group elements are represented in a compressed form by SLPs, which generalizes the classical NP-completeness result of the integer knapsack problem. We also prove general transfer results: NP-membership of th...

It is shown that the positive first-order theory of an HNN-extension G = 〈H, t; tat = φ(a)(a ∈ A)〉 can be reduced to the existential firstorder theory of G provided that A and B are proper subgroups of the base group H with A∩B finite. For an amalgamated free product G = H ∗A J , we show that the positive first-order theory of G can be reduced to the existential first-order theory of G in case ...

The category of crossed complexes gives an algebraic model of CW -complexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating non-abelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions...

Linear and projective boundaries of Cayley graphs were introduced in [6] as quasi-isometry invariant boundaries of finitely generated groups. They consist of forward orbits g D 1g W i 2 No, or orbits g D 1g W i 2 Zo, respectively, of non-torsion elements g of the group G, where ‘sufficiently close’ (forward) orbits become identified, together with a metric bounded by 1. We show that for all fin...

Abstract We show that the commensurator of any finitely generated abelian subgroup H in a biautomatic group centralises finite-index . deduce CAT(0) groups introduced by Leary and Minasyan (Commensurating HNN-extensions: non-positive curvature biautomaticity, Geom Topol 25 :1819–1860, 2021) are either or cannot arise as subgroups groups, answering question posed generalising an analogous result...

Abstract We study the quasi-isometric rigidity of a large family finitely generated groups that split as graphs with virtually free vertex and two-ended edge groups. Let G be group is one-ended, hyperbolic relative to abelian subgroups, has JSJ decomposition over subgroups containing only are not quadratically hanging. Our main result any abstractly commensurable . In particular, our applies ce...