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Let F=\langle a,b\rangle be the free group on two generators, and let \phi:F \to F be the endomorphism defined on generators by \phi(a)=ab and \phi(b)=ba. We define Sapir’s group C to be the ascending HNN extension

F*_\phi:=\langle a,b,t\; | \; a^t=ab,b^t=ba\rangle

This group was studied by Crisp-Sageev-Sapir in the context of their work on right-angled Artin groups, and independently by Feighn (according to Mark Sapir); both sought (unsuccessfully) to determine whether C contains a subgroup isomorphic to the fundamental group of a closed, oriented surface of genus at least 2. Sapir has conjectured in personal communication that C does not contain a surface subgroup, and explicitly posed this question as Problem 8.1 in his problem list.

After three years of thinking about this question on and off, Alden Walker and I have recently succeeded in finding a surface subgroup of C, and it is the purpose of this blog post to describe this surface, how it was found, and some related observations. By pushing the technique further, Alden and I managed to prove that for a fixed free group F of finite rank, and for a random endomorphism \phi of length n (i.e. one taking the generators to random words of length n), the associated HNN extension contains a closed surface subgroup with probability going to 1 as n \to \infty. This result is part of a larger project which we expect to post to the arXiv soon.

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