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Algorithmic and statistical properties of filling elements of a free group, and quantitative residual properties of gamma-limit groups

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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: Kapovich, Ilya
Doctoral Committee Chair(s): Leininger, Christopher J.
Doctoral Committee Member(s): Kapovich, Ilya; 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
 

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