Date of Degree

6-2023

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Olga Kharlampovich

Committee Members

Ilya Kapovich

Vladimir Shpilrain

Subject Categories

Algebra | Geometry and Topology | Mathematics

Keywords

retractions, fully residually free, elevations, precovers, GICE, ICE

Abstract

We show that for any finitely generated non-abelian subgroup H of a limit group L, there exists a finite-index subgroup K which is fully residually H. This generalizes the result of Wilton that limit groups admit local retractions. 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 results of Louder, McReynolds, and Patel for free and surface groups.

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