742 research outputs found
Detecting free splittings in relatively hyperbolic groups
We describe an algorithm which determines whether or not a group which is
hyperbolic relative to abelian groups admits a nontrivial splitting over a
finite group.Comment: 15 pages. Version 2 is 17 pages, edited in light of referee's
comments. To appear in Transactions of the AM
Conjugacy classes of solutions to equations and inequations over hyperbolic groups
We study conjugacy classes of solutions to systems of equations and
inequations over torsion-free hyperbolic groups, and describe an algorithm to
recognize whether or not there are finitely many conjugacy classes of solutions
to such a system. The class of immutable subgroups of hyperbolic groups is
introduced, which is fundamental to the study of equations in this context. We
apply our results to enumerate the immutable subgroups of a torsion-free
hyperbolic group.Comment: 28 pages; referee's comments incorporated; to appear in the Journal
of Topolog
Dehn filling in relatively hyperbolic groups
We introduce a number of new tools for the study of relatively hyperbolic
groups. First, given a relatively hyperbolic group G, we construct a nice
combinatorial Gromov hyperbolic model space acted on properly by G, which
reflects the relative hyperbolicity of G in many natural ways. Second, we
construct two useful bicombings on this space. The first of these, "preferred
paths", is combinatorial in nature and allows us to define the second, a
relatively hyperbolic version of a construction of Mineyev.
As an application, we prove a group-theoretic analog of the Gromov-Thurston
2\pi Theorem in the context of relatively hyperbolic groups.Comment: 83 pages. v2: An improved version of preferred paths is given, in
which preferred triangles no longer need feet. v3: Fixed several small errors
pointed out by the referee, and repaired several broken figures. v4:
corrected definition 2.38. This is very close to the published versio
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