We define the notion of acylindrical graph of groups of a group. We bound the combinatorics of these graphs of groups for f.g. freely indecomposable groups. Our arguments imply the finiteness of acylindrical surfaces in closed 3-manifolds [Ha], finiteness of isomorphism classes of small splittings of (torsion-free) freely indecomposable hyperbolic groups as well as finiteness results for small splittings of f.g. Kleinian and semisimple discrete groups acting on non-positively curved simply connected manifolds. In order to get our accessibility for f.g. groups we generalize parts of Rips' analysis of stable actions of f.p. groups on real trees to f.g. groups. The concepts we present play an essential role in constructing the canonical JSJ decomposition ([Se1],[Ri-Se2]), in obtaining the Hopf property for hyperbolic groups [Se2], and in our study of sets of solutions to equations in a free group [Se3].
KeywordsManifold Free Group Essential Role Isomorphism Class Discrete Group
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