Quasiconvex Subgroups and Nets in Hyperbolic Groups

Author: Mack, Thomas Patrick

Year: 2006

Degree: Dissertation (Ph.D.)

Advisor: Calegari, Danny C.

Committee Members: Calegari, Danny C.; Oh, Hee; Aschbacher, Michael; Dunfield, Nathan M.

Option: Mathematics

DOI: 10.7907/35GG-W072

Abstract

Consider a hyperbolic group G and a quasiconvex subgroup H of G with [G:H] infinite. We construct a set-theoretic section s:G/H -> G of the quotient map (of sets) G -> G/H such that s(G/H) is a net in G; that is, any element of G is a bounded distance from s(G/H). This set arises naturally as a set of points minimizing word-length in each fixed coset gH. The left action of G on G/H induces an action on s(G/H), which we use to prove that H contains no infinite subgroups normal in G.

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