The structure of groups with a quasiconvex hierarchy

The structure of groups with a quasiconvex hierarchy

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Daniel Wise, McGill University
Fine Hall 314

We prove that hyperbolic groups with a quasiconvex hierarchy are virtually subgroups of graph groups. Our focus is on "special cube complexes" which are nonpositively curved cube complexes that behave like "high dimensional graphs" and are closely related to graph groups. The main result illuminates the structure of a group by showing that it is "virtually special", and this yields the separability of the quasiconvex subgroups of the groups we study.As an application, we resolve Baumslag's conjecture on the residual finiteness of one-relator groups with torsion. Another application shows that generic haken hyperbolic 3-manifolds have "virtually special" fundamental group. Since graph groups are residually finite rational solvable, combined with Agol's virtual fibering criterion, this proves that finite volume haken hyperbolic 3-manifolds are virtually fibered.