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Abstract We define the notion of accessibility for a pro- p group. We prove that finitely generated pro- p groups are accessible given a bound on the size of their finite subgroups. We then construct a finitely generated inaccessible pro- p group, and also a finitely generated inaccessible discrete group which is residually p -finite. [ABSTRACT FROM AUTHOR]

We discuss a partial normalisation of a finite graph of finite groups $$(\Gamma (-), X)$$ which leaves invariant the fundamental group. In conjunction with an easy graph-theoretic result, this provides a flexible and rather useful tool in the study of finitely generated virtually free groups. Applications discussed here include: (1) an important inequality for the number of edges in a Stallings decomposition $$\Gamma \cong \pi _1(\Gamma (-), X)$$ of a finitely generated virtually free group, (2) the proof of equivalence of a number of conditions for such a group to be 'large', as well as (3) the classification up to isomorphism of virtually free groups of (free) rank 2. We also discuss some number-theoretic consequences of the last result. [ABSTRACT FROM AUTHOR]

We define a Levi category to be a weakly orthogonal category equipped with a suitable length functor and prove two main theorems about them. First, skeletal cancellative Levi categories are precisely the categorical versions of graphs of groups with a given orientation. Second, the universal groupoid of a skeletal cancellative Levi category is the fundamental groupoid of the corresponding graph of groups. These two results can be viewed as a co-ordinate-free refinement of a classical theorem of Philip Higgins. [ABSTRACT FROM AUTHOR]

Let G split as a finite graph of free abelian groups with cyclic edge groups. We characterize when G acts freely on a CAT(0) cube complex. We show that if G acts properly and semi-simply on a CAT(0) space, then G acts freely on a CAT(0) cube complex. [ABSTRACT FROM AUTHOR]

A group G is a v group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We describe the compatibility JSJ decomposition over abelian groups. We prove that in general this decomposition is not algorithmically computable. [ABSTRACT FROM AUTHOR]

Let A be a symmetrizable affine or hyperbolic generalized Cartan matrix. Let G be a locally compact Kac-Moody group associated to A over a finite field 픽q. We suppose that G has type ∞, that is, the Weyl group W of G is a free product of ℤ/2ℤ's. This includes all locally compact Kac-Moody groups of rank 2 and three possible locally compact rank 3 Kac-Moody groups of noncompact hyperbolic type. For every prime power q, we give a sufficient condition for the rank 2 Kac-Moody group G to contain a cocompact lattice $\Gamma \cong M_q \ast_{M_q\cap \widetilde{M_q}}\widetilde{M_q}$ with quotient a simplex, and we show that this condition is satisfied when q = 2s. If further Mq and $\widetilde{M_q}$ are abelian, we give a method for constructing an infinite descending chain of cocompact lattices ... Γ3 ≤ Γ2 ≤ Γ1 ≤ Γ. This allows us to characterize each of the quotient graphs of groups Γi\\X, the presentations of the Γi and their covolumes, where X is the Tits building of G, a homogeneous tree. Our approach is to extend coverings of edge-indexed graphs to covering morphisms of graphs of groups with abelian groupings. This method is not specific to cocompact lattices in Kac-Moody groups and may be used to produce chains of subgroups acting on trees in a general setting. It follows that the lattices constructed in the rank 2 Kac-Moody group have the Haagerup property. When q = 2 and rank(G) = 3 we show that G contains a cocompact lattice Γ′1 that acts discretely and cocompactly on a simplicial tree $\mathcal{X}$. The tree $\mathcal{X}$ is naturally embedded in the Tits building X of G, a rank 3 hyperbolic building. Moreover Γ′1 ≤ Λ′ for a non-discrete subgroup Λ′ ≤ G whose quotient Λ′ \ X is equal to G\X. Using the action of Γ′1 on $\mathcal{X}$ we construct an infinite descending chain of cocompact lattices ...Γ′3 ≤ Γ′2 ≤ Γ′1 in G. We also determine the quotient graphs of groups $\Gamma_i'\backslash\!\backslash\mathcal{X}$, the presentations of the Γ′i and their covolumes. [ABSTRACT FROM AUTHOR]

Abstract: It is shown that the fundamental groups of certain non-positively curved 2-complexes have the property that their quasiconvex subgroups are the intersections of finite index subgroups. As a consequence, every geometrically finite subgroup of the figure 8 knot group is the intersection of finite index subgroups. The same result holds for many other prime alternating link groups. [Copyright &y& Elsevier]

We show that if the fundamental group of a finite graph of groups with finitely generated Abelian vertex groups is subgroup separable, then it is linear. [ABSTRACT FROM AUTHOR]

We introduce a combinatorial version of Stallings–Bestvina–Feighn–Dunwoody folding sequences. We then show how they are useful in analyzing the solvability of the uniform subgroup membership problem for fundamental groups of graphs of groups. Applications include coherent right-angled Artin groups and coherent solvable groups. [ABSTRACT FROM AUTHOR]

We give sufficient conditions for the fundamental groupoid of a graph of groups to be automatic (resp. asynchronously automatic). These conditions are similar to those in [1] for a free product with amalgamation to be automatic (resp. asynchronously automatic). [ABSTRACT FROM AUTHOR]

Gouliang Yu has introduced a property of discrete metric spaces and groups called property A which implies the coarse Baum–Connes Conjecture and hence the Novikov Higher Signature Conjecture. In this paper we extend a result of Jean-Louis Tu to conclude that a group acting by isometries on a metric space with finite asymptotic dimension whose d-stabilizers have property A, also has property A. As a result, we conclude a theorem of Tu, according to which, a fundamental group of a finite graph of groups whose vertices have property A also has property A. [Copyright &y& Elsevier]