Automorphisms and endomorphisms of lacunary hyperbolic groups

  • Rémi Coulon

    Université de Rennes 1, France
  • Vincent Guirardel

    Université de Rennes 1, France
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Abstract

In this article we study automorphisms and endomorphisms of lacunary hyperbolic groups. We prove that every lacunary hyperbolic group is Hopfian, answering a question by Henry Wilton. In addition, we show that if a lacunary hyperbolic group has the fixed point property for actions on -trees, then it is co-Hopfian and its outer automorphism group is locally finite. We also construct lacunary hyperbolic groups whose automorphism group is infinite, locally finite, and contains any locally finite group given in advance.

Cite this article

Rémi Coulon, Vincent Guirardel, Automorphisms and endomorphisms of lacunary hyperbolic groups. Groups Geom. Dyn. 13 (2019), no. 1, pp. 131–148

DOI 10.4171/GGD/488