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# 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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## Permalink

https://hdl.handle.net/2142/24044

## 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
- Permalink
- http://hdl.handle.net/2142/24044
- Copyright and License Information
- Copyright 2011 Brent B. Solie

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