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Title:  Algorithmic and statistical properties of filling elements of a free group, and quantitative residual properties of gammalimit 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 UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  filling element
filling subgroup free group CullerVogtmann 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 Rtree. 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 nonfilling 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 ArzhantsevaOl’shanskii, refining a result of Kapovich and Lustig. Let Γ be a fixed hyperbolic group. The Γlimit groups of Sela are exactly the finitely generated, fully residually Γ groups. We give a new invariant of Γlimit groups called Γdiscriminating complexity and show that the Γdiscriminating complexity of any Γlimit group is asymptotically dominated by a polynomial. Our proof relies on an embedding theorem of KharlampovichMyasnikov which states that a Γlimit 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 Gdiscriminating complexity of a rank n extension of a cyclic centralizer of G is asymptotically dominated by a polynomial of degree n. 
Issue Date:  20110525 
URI:  http://hdl.handle.net/2142/24044 
Rights Information:  Copyright 2011 Brent B. Solie 
Date Available in IDEALS:  20110525 
Date Deposited:  201105 
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Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois 
Dissertations and Theses  Mathematics
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