2026-09-21T00:29:31Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/24044 2023-07-10 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Kapovitch, Ilia Leininger, Christopher J. Kapovitch, Ilia Mineyev, Igor Robinson, Derek J.S. Solie, Brent B. 2011-05-25T15:01:24Z 2011-05-25T15:01:24Z 2011-05-25T15:01:24Z 2011-05 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of a rank n extension of a cyclic centralizer of G is asymptotically dominated by a polynomial of degree n. Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2011-04-15T22:34:29Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Solie_Brent.pdf: 1740211 bytes, checksum: d536e219d0a66d1effa905f86c974501 (MD5) Made available in DSpace on 2011-05-25T15:01:24Z (GMT). No. of bitstreams: 2 Solie_Brent.pdf: 1741468 bytes, checksum: 23721f2675ef74f5e69359801e84e3ff (MD5) license.txt: 4058 bytes, checksum: 49380124cde7779e73d4eff9f742b531 (MD5) http://hdl.handle.net/2142/24044 en Copyright 2011 Brent B. Solie filling element filling subgroup free group Culler-Vogtmann outer space groups acting on trees genericity limit groups relatively hyperbolic groups hyperbolic geometry residual properties Algorithmic and statistical properties of filling elements of a free group, and quantitative residual properties of gamma-limit groups Mathematics 1257 Mathematics 0439 University of Illinois at Urbana-Champaign Dissertation Ph.D. PHD:Mathematics -UIUC 10KS0439PHD