Surface Groups Within Baumslag Doubles

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 ∈ F . Furthermore, a hyperbolic Baumslag double G contains a non-orientable surface group of genus 4 if and only if W = X2Y 2 for some X,Y ∈ F . G can contain no non-orientable surface group of smaller genus.