References & Citations
Mathematics > Group Theory
Title: Quantifying separability in limit groups via representations
(Submitted on 7 Mar 2023 (v1), last revised 11 Apr 2023 (this version, v2))
Abstract: We show that for any finitely generated subgroup $H$ of a limit group $L$ there exists a finite-index subgroup $K$ containing $H$, such that $K$ is a subgroup of a group obtained from $H$ by a series of extensions of centralizers and free products with $\mathbb Z$. If $H$ is non-abelian, the $K$ is fully residually $H$. We also show that for any finitely generated subgroup of a limit group, there is a finite-dimensional representation of the limit group which separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate a finitely generated subgroup in a limit group. This generalizes the results of Louder, McReynolds and Patel. Another corollary is that a hyperbolic limit group satisfies the Geometric Hanna Neumann conjecture.
Submission history
From: Olga Kharlampovich [view email][v1] Tue, 7 Mar 2023 04:21:51 GMT (414kb,D)
[v2] Tue, 11 Apr 2023 14:03:50 GMT (6001kb,D)
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