On surface subgroups of doubles of free groups

Abstract

We give several sufficient conditions for a double of a free group along a cyclic subgroup to contain a surface subgroup.

Reflections on some aspects of infinite groups

“…Gromov (see [29,33]) conjectured that a one-ended word hyperbolic group must contain a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. The general question is still open.…”

“…The general question is still open. Recent work by Gordon and Wilton [29] and Kim and Wilton [33] provide su cient conditions for hyperbolic surface groups to be embedded in a Baumslag double . The work of Gordon and Wilton uses group cohomology and 3-manifold theory while that of Kim and Wilton proceeds by realizing a Baumslag double as the fundamental group of a non-positively curved square complex.…”

Surface Subgroups of Graph Products of Groups

“…Question 2 has been answered for only a few cases, all affirmatively. These include graphs of free groups with cyclic edge groups with nontrivial second rational homology [5], doubles of rank-two free groups symmetrically amalgamated along cyclic edge groups [19,34,33], and most remarkably, the fundamental groups of closed hyperbolic 3-manifolds [29]. We note that these groups are all virtually cocompact special in the sense that each one is virtually the fundamental group of a compact special cube complex [24,25] a ; see Definition 10.…”

Virtually geometric words and Whitehead’s algorithm

“…A 3-manifold argument based on Dehn's Lemma [10,Lemma 20] then shows that if D n (w) is one-ended, it contains a closed surface subgroup.…”

“…Gordon and Wilton gave some examples of virtually geometric but non-geometric words in [10], and asked the following: Thus a positive answer to Question 1.6 for words in [F, F ] would have given an alternate proof of the rationality of scl in free groups, which is the main theorem of [5]. In fact, the answer to Question 1.6 is negative even in this more restricted setting (see Corollary 4.3).…”