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.

We obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of n-generated one-ended subgroups.2000 Mathematics Subject Classification. 20F67.

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We give a detailed account of Zlil Sela's construction of Makanin-Razborov diagrams describing Hom(G, Γ) where G is a finitely generated group and Γ is a hyperbolic group. We also deal with the case where Γ has torsion. Les diagrammes Makanin-Razborov pour les groupes hyperboliques Résumé Nous proposons une présentation détaillée de la construction des diagrammes de Makanin-Razborov de Zlil Sela qui décrivent Hom(G, Γ) pour un groupe G de type fini et un groupe hyperbolique Γ. De plus, nous traitons le cas où Γ est un groupe ayant de la torsion.

A Coxeter system is called skew-angled if its Coxeter matrix contains no entry equal to 2. In this paper we prove rigidity results for skew-angled Coxeter groups. As a consequence of our results we obtain that skew-angled Coxeter groups are rigid up to diagram twisting.3. If S has at least 3 elements and if GðW ; SÞ is edge-connected, then there exists w A W such that S w ¼ S.Reformulating statement (1) of the main theorem in the language of [3] we get the following corollary. It implies that Conjecture 8.1 in [3] holds in the skew-angled case.Corollary A. Skew-angled Coxeter systems are reflection-rigid up to diagram twisting.Remark 1. Let ðW ; SÞ be a Coxeter system and let s; t A S be two reflections corresponding to the vertices which are on a bridge of GðW ; SÞ. If there is a non-trivial reflection-preserving outer automorphism a of hs; ti (like for instance in the case where st has order 5), then it has an extension to a reflection-preserving automorphism b of W and bðSÞ is not twist equivalent to S because twistings are 'anglepreserving'.If ðW 1 ; S 1 Þ and ðW 2 ; S 2 Þ are both skew-angled Coxeter systems then any isomorphism f : W 1 ! W 2 maps reflections onto reflections since the parabolic dihedral subgroups are the maximal finite subgroups and since any automorphism of a dihedral group D 2n with n d 3 maps reflections onto reflections. The theorem therefore gives a solution to the isomorphism problem for the class of skew-angled Coxeter groups. This can be rephrased as follows:Corollary B. Given two fundamental sets S; S 0 in a Coxeter group W such that ðW ; SÞ and ðW ; S 0 Þ are skew-angled, then GðW ; SÞ and GðW ; S 0 Þ are twist equivalent.As there are no spikes in an edge-connected graph, Part 3 of the main theorem and the previous theorem have the following consequence:Corollary C. Skew-angled Coxeter systems whose diagram has no spike which is labelled by twice an odd number are rigid up to diagram twisting (in the sense of [3]);