Description
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 
Files in this item
Files  Description  Format 

application/pdf Solie_Brent.pdf (2Mb)  (no description provided) 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois 
Dissertations and Theses  Mathematics
Item Statistics
 Total Downloads: 131
 Downloads this Month: 0
 Downloads Today: 0