Surface Groups Within Baumslag Doubles
Copyright 2011 by Cambridge University Press. Original published version can be found at DOI: 10.1017/S0013091509001102
A conjecture of Gromov states that a one-ended word-hyperbolic group must contain a subgroup that is isomorphic to the fundamental group of a closed hyperbolic surface. Recent papers by Gordon and Wilton and by Kim and Wilton give sufficient conditions for hyperbolic surface groups to be embedded in a hyperbolic Baumslag double G. Using Nielsen cancellation methods based on techniques from previous work by the second author, we prove that a hyperbolic orientable surface group of genus 2 is embedded in a hyperbolic Baumslag double if and only if the amalgamated word W is a commutator: that is, W = [U, V] for some elements U, V is an element of F. Furthermore, a hyperbolic Baumslag double G contains a non-orientable surface group of genus 4 if and only if W = X(2)Y(2) for some X, V is an element of F. G can contain no non-orientable surface group of smaller genus.
Benjamin Fine and Gerhard Rosenberger. "Surface Groups Within Baumslag Doubles" Proceedings of the Edinburgh Mathematical Society 54.Part 1 (2011).
Available at: http://works.bepress.com/benjamin_fine/1