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Title: | Algorithmic and statistical properties of filling elements of a free group, and quantitative residual properties of gamma-limit groups |
Author(s): | Solie, Brent B. |
Director of Research: | Kapovitch, Ilia |
Doctoral Committee Chair(s): | Leininger, Christopher J. |
Doctoral Committee Member(s): | Kapovitch, Ilia; Mineyev, Igor; Robinson, Derek J.S. |
Department / Program: | Mathematics |
Discipline: | Mathematics |
Degree Granting Institution: | University of Illinois at Urbana-Champaign |
Degree: | Ph.D. |
Genre: | Dissertation |
Subject(s): | filling element
filling subgroup free group Culler-Vogtmann outer space groups acting on trees genericity limit groups relatively hyperbolic groups hyperbolic geometry residual properties |
Abstract: | A filling subgroup of a finitely generated free group F(X) is a subgroup which does not fix a point in any very small action free action on an R-tree. For the free group of rank two, we construct a combinatorial algorithm to determine whether or not a given finitely generated subgroup is filling. In higher ranks, we discuss two types of non-filling subgroups: those contained in loop vertex subgroups and those contained in segment vertex subgroups. We construct a combinatorial algorithm to determine whether or not a given finitely generated subgroup is contained in a segment vertex subgroup. We further give a combinatorial algorithm which identifies a certain kind of subgroup contained in a loop vertex subgroup. Finally, we show that the set of filling elements of F(X) is exponentially generic in the sense of Arzhantseva-Ol’shanskii, refining a result of Kapovich and Lustig. Let Γ be a fixed hyperbolic group. The Γ-limit groups of Sela are exactly the finitely generated, fully residually Γ groups. We give a new invariant of Γ-limit groups called Γ-discriminating complexity and show that the Γ-discriminating complexity of any Γ-limit group is asymptotically dominated by a polynomial. Our proof relies on an embedding theorem of Kharlampovich-Myasnikov which states that a Γ-limit group embeds in an iterated extension of centralizers over Γ.The result then follows from our proof that if G is an iterated extension of centralizers over Γ, the G-discriminating complexity of a rank n extension of a cyclic centralizer of G is asymptotically dominated by a polynomial of degree n. |
Issue Date: | 2011-05-25 |
URI: | http://hdl.handle.net/2142/24044 |
Rights Information: | Copyright 2011 Brent B. Solie |
Date Available in IDEALS: | 2011-05-25 |
Date Deposited: | 2011-05 |
This item appears in the following Collection(s)
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Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois -
Dissertations and Theses - Mathematics