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Treelike structures arising from continua and convergence groups. (English) Zbl 0961.20034
Mem. Am. Math. Soc. 662, 86 p. (1999).
The author introduces a notion of treelike structure, which he calls “pretree”, which encompasses most of the existing notions of trees (simplicial trees, \(\Lambda\)-trees, protrees, pseudo-trees, etc.). The theory that he develops here unifies in some sense the existing theories. A theme in this paper is the applications of pretrees to the topology of continua. Here, a “continuum” is a connected compact Hausdorff topological space. The principal construction in this paper associates to a continuum a natural quotient which is a dendrite. The author obtains a result stating that a general condition on a convergence group (in the sense of Gehring and Martin) acting on a continuum implies that every global cut point is a parabolic fixed point. This is related to a conjecture of Bestvina and Mess stating that the boundary of a one-ended hyperbolic group is locally connected. Partial results in this direction had been obtained by several authors. The author discusses in this paper several results in this direction. The paper contains in particular several new results on hyperbolic groups. The author proves that if \(\Gamma\) is a one-ended finitely generated group which admits a minimal convergence action on a continuum \(M\) and if \(M\) has a cut-point which is not a parabolic fixed point, then the quotient is not a point. As a consequence, he proves that if the boundary of a one-ended hyperbolic group contains global cut-points, then the group splits over a virtually cyclic subgroup.

20F65 Geometric group theory
20E08 Groups acting on trees
20F67 Hyperbolic groups and nonpositively curved groups
54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
57M60 Group actions on manifolds and cell complexes in low dimensions
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