We re-cast in a more combinatorial and computational form the topological approach of J. Stallings to the study of subgroups of free groups.  2002 Elsevier Science (USA)

We give a precise definition of "generic-case complexity" and show that for a very large class of finitely generated groups the classical decision problems of group theory -the word, conjugacy and membership problems -all have linear-time generic-case complexity. We prove such theorems by using the theory of random walks on regular graphs.

We prove that Whitehead's algorithm for solving the automorphism problem in a fixed free group F k has strongly linear time generic-case complexity. This is done by showing that the "hard" part of the algorithm terminates in linear time on an exponentially generic set of input pairs. We then apply these results to one-relator groups. We obtain a Mostow-type isomorphism rigidity result for random one-relator groups: If two such groups are isomorphic then their Cayley graphs on the given generating sets are isometric. Although no nontrivial examples were previously known, we prove that one-relator groups are generically complete groups, that is, they have trivial center and trivial outer automorphism group. We also prove that the stabilizers of generic elements of F k in Aut(F k ) are cyclic groups generated by inner automorphisms and that Aut(F k )-orbits are uniformly small in the sense of their growth entropy. We further prove that the number I k (n) of isomorphism types of k-generator one-relator groups with defining relators of length n satisfies c 1 n (2k − 1)where c 1 , c 2 are positive constants depending on k but not on n. Thus I k (n) grows in essentially the same manner as the number of cyclic words of length n.

1805Geometric intersection number and analogues of the curve complex for free groups ILYA KAPOVICH MARTIN LUSTIGFor the free group F N of finite rank N 2 we construct a canonical Bonahon-type, continuous and Out.F N /-invariant geometric intersection formHere cv.F N / is the closure of unprojectivized Culler-Vogtmann Outer space cv.F N / in the equivariant Gromov-Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that cv.F N / consists of all very small minimal isometric actions of F N on R-trees. The projectivization of cv.F N / provides a free group analogue of Thurston's compactification of Teichmüller space.As an application, using the intersection graph determined by the intersection form, we show that several natural analogues of the curve complex in the free group context have infinite diameter.

Abstract. 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.

In this paper we survey the known results about boundaries of word-hyperbolic groups.1991 Mathematics Subject Classification. [. 1Example 2.2. Any geodesic metric space of finite diameter D is D-hyperbolic. An R-tree (e.g. the real line R or a simplicial tree) is 0-hyperbolic. The standard hyperbolic plane H 2 is log( √ 2 + 1)-hyperbolic. If M is a closed Riemannian manifold of negative sectional curvature then the universal covering space of M (with the induced metric) is hyperbolic.Another basic example of hyperbolic spaces is provided by the so-called CAT (k)-spaces, where k < 0. We recall their definition:Definition 2.3. Let (X, d) be a geodesic metric space and let k ≤ 0. We say that (X, d) is a CAT (k)-space if the following holds.

Abstract. Let F be a free group of rank N ≥ 2, let µ be a geodesic current on F and let T be an R-tree with a very small isometric action of F . We prove that the geometric intersection number T, µ is equal to zero if and only if the support of µ is contained in the dual algebraic lamination L 2 (T ) of T . Applying this result, we obtain a generalization of a theorem of Francaviglia regarding length spectrum compactness for currents with full support. We use the main result to obtain "unique ergodicity" type properties for the attracting and repelling fixed points of atoroidal iwip elements of Out(F ) when acting both on the compactified outer Space and on the projectivized space of currents. We also show that the sum of the translation length functions of any two "sufficiently transverse" very small F -trees is bilipschitz equivalent to the translation length function of an interior point of the outer space. As another application, we define the notion of a filling element in F and prove that filling elements are "nearly generic" in F . We also apply our results to the notion of bounded translation equivalence in free groups.

Abstract. Given a free group