Splittings and automorphisms of relatively hyperbolic groups

We study automorphisms of a relatively hyperbolic group G. When G is one-ended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GL_n(Z) fixing certain basis elements. When more general parabolic groups are allowed, these subgroups of GL_n(Z) have to be replaced by McCool groups: automorphisms of parabolic groups acting trivially (i.e. by conjugation) on certain subgroups. Given a malnormal quasiconvex subgroup P of a hyperbolic group G, we view G as hyperbolic relative to P and we apply the previous analysis to describe the group Out(P to G) of automorphisms of P that extend to G: it is virtually a McCool group. If Out(P to G) is infinite, then P is a vertex group in a splitting of G. If P is torsion-free, then Out(P to G) is of type VF, in particular finitely presented. We also determine when Out(G) is infinite, for G relatively hyperbolic. The interesting case is when G is infinitely-ended and has torsion. When G is hyperbolic, we show that Out(G) is infinite if and only if G splits over a maximal virtually cyclic subgroup with infinite center. In general we show that infiniteness of Out(G) comes from the existence of a splitting with infinitely many twists, or having a vertex group that is maximal parabolic with infinitely many automorphisms acting trivially on incident edge groups.Comment: Minor modifications. To appear in Geometry Groups and Dynamic

The automorphism group of accessible groups

In this article, we study the outer automorphism group of a group G decomposed as a finite graph of group with finite edge groups and finitely generated vertex groups with at most one end. We show that Out(G) is essentially obtained by taking extensions of relative automorphism groups of vertex groups, groups of Dehn twists and groups of automorphisms of free products. We apply this description and obtain a criterion for Out(G) to be finitely presented, as well as a necessary and sufficient condition for Out(G) to be finite. Consequences for hyperbolic groups are discussed.Comment: 18 pages, 3 figures. Section 4 rewritten and corrected, added reference

Quasiconvex Subgroups and Nets in Hyperbolic Groups

Consider a hyperbolic group G and a quasiconvex subgroup H of infinite index. We construct a set-theoretic section s of the quotient map (of sets) from G to G/H such that s(G/H) is a net in G; that is, any element of G is a bounded distance from s(G/H). This section arises naturally as a set of points minimizing word-length in each fixed coset gH. The left action of G on G/H induces an action on s(G/H), which we use to prove that H contains no infinite subgroups normal in G.Comment: 15 pages, 1 figure; v3: Replaced another typo; v2: Replaced minor typo in abstrac

On the difficulty of presenting finitely presentable groups

We exhibit classes of groups in which the word problem is uniformly solvable but in which there is no algorithm that can compute finite presentations for finitely presentable subgroups. Direct products of hyperbolic groups, groups of integer matrices, and right-angled Coxeter groups form such classes. We discuss related classes of groups in which there does exist an algorithm to compute finite presentations for finitely presentable subgroups. We also construct a finitely presented group that has a polynomial Dehn function but in which there is no algorithm to compute the first Betti number of the finitely presentable subgroups.Comment: Final version. To appear in GGD volume dedicated to Fritz Grunewal

Residual properties of 3-manifold groups I: Fibered and hyperbolic 3-manifolds

Let $p$ be a prime. In this paper, we classify the geometric 3-manifolds whose fundamental groups are virtually residually $p$. Let $M=M^3$ be a virtually fibered 3-manifold. It is well-known that $G=\pi_1(M)$ is residually solvable and even residually finite solvable. We prove that $G$ is always virtually residually $p$. Using recent work of Wise, we prove that every hyperbolic 3-manifold is either closed or virtually fibered and hence has a virtually residually $p$ fundamental group. We give some generalizations to pro-$p$ completions of groups, mapping class groups, residually torsion-free nilpotent 3-manifold groups and central extensions of residually $p$ groups.Comment: 25 pages. Complete rewrit