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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.