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

Solie, Brent B.

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Description

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.

Issue Date

2011-05-25T15:01:24Z

Director of Research (if dissertation) or Advisor (if thesis)

Kapovitch, Ilia

Doctoral Committee Chair(s)

Leininger, Christopher J.

Committee Member(s)

• Kapovitch, Ilia
• Mineyev, Igor
• Robinson, Derek J.S.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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.

Graduation Semester

2011-05

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Copyright and License Information

Copyright 2011 Brent B. Solie

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