{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/24044"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/24044","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Algorithmic and statistical properties of filling elements of a free group, and quantitative residual properties of gamma-limit groups","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Solie, Brent B."],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Kapovitch, Ilia","Leininger, Christopher J.","Mineyev, Igor","Robinson, Derek J.S."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-25T15:01:24Z","date_published":"2011-05-25T15:01:24Z","updated_at":"2026-07-22T22:25:23Z","subjects":["filling element","filling subgroup","free group","Culler-Vogtmann outer space","groups acting on trees","genericity","limit groups","relatively hyperbolic groups","hyperbolic geometry","residual properties"],"languages":["en"],"rights":["Copyright 2011 Brent B. Solie"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/24044","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Kapovitch, Ilia","Leininger, Christopher J.","Mineyev, Igor","Robinson, Derek J.S."]},{"key":"dc:creator","label":"Author","values":["Solie, Brent B."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-25T15:01:24Z","2011-05"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["filling element","filling subgroup","free group","Culler-Vogtmann outer space","groups acting on trees","genericity","limit groups","relatively hyperbolic groups","hyperbolic geometry","residual properties"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2011 Brent B. Solie"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/24044"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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)"]},{"key":"dc:title","label":"Title","values":["Algorithmic and statistical properties of filling elements of a free group, and quantitative residual properties of gamma-limit groups"]}]}],"canonical_facts":{"dc:contributor":["Kapovitch, Ilia","Leininger, Christopher J.","Mineyev, Igor","Robinson, Derek J.S."],"dc:creator":["Solie, Brent B."],"dc:date":["2011-05-25T15:01:24Z","2011-05"],"dc:description":["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)"],"dc:identifier":["http://hdl.handle.net/2142/24044"],"dc:language":["en"],"dc:rights":["Copyright 2011 Brent B. Solie"],"dc:subject":["filling element","filling subgroup","free group","Culler-Vogtmann outer space","groups acting on trees","genericity","limit groups","relatively hyperbolic groups","hyperbolic geometry","residual properties"],"dc:title":["Algorithmic and statistical properties of filling elements of a free group, and quantitative residual properties of gamma-limit groups"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:23Z"}