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be a group hyperbolic relative to a finite collection of subgroups
be the family of subgroups consisting of all the conjugates of subgroups in
, all their subgroups, and all finite subgroups. Then there is a cocompact model for
. This result was known in the torsion-free case. In the presence of torsion, a new approach was necessary. Our method is to exploit the notion of dismantlability. A number of sample applications are discussed.
We prove that for an action of a finite group G on a systolic complex X there exists a G–invariant subcomplex of X of diameter ≤5. For 7–systolic locally finite complexes we prove there is a fixed point for the action of any finite G. This implies that free products with amalgamation (and HNN extensions) of 7–systolic groups over finite subgroups are also 7–systolic.
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