{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/68184"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/68184","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Solutions of Quadratic Equations in Small Cancellation Groups","abstract":"The main results are classifications of the solutions of quadratic equations in small cancellation quotients of free groups and free products. Paul E. Schupp has obtained similar results for a restricted class of quadratic equations (those without constants). Some of the methods used here were previously employed by Leo P. Comerford and Charles C. Edmunds in their study of algorithmic problems on the solutions of quadratic equations. The principal tools are the construction of cancellation diagrams on compact surfaces with boundary and an analysis of the co-initial graphs of quadratic words.","abstract_html":"The main results are classifications of the solutions of quadratic equations in small cancellation quotients of free groups and free products. Paul E. Schupp has obtained similar results for a restricted class of quadratic equations (those without constants). Some of the methods used here were previously employed by Leo P. Comerford and Charles C. Edmunds in their study of algorithmic problems on the solutions of quadratic equations. The principal tools are the construction of cancellation diagrams on compact surfaces with boundary and an analysis of the co-initial graphs of quadratic words.","abstract_has_math":false,"creators":["Anderson, Claude Wilson, III"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-14T13:09:50Z","date_published":"2014-12-14T13:09:50Z","updated_at":"2026-07-22T22:25:58Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8203393"],"render_values":[{"text":"(UMI)AAI8203393","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/68184","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Anderson, Claude Wilson, III"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-14T13:09:50Z","10000-01-01","1981"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"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":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/68184","(UMI)AAI8203393"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The main results are classifications of the solutions of quadratic equations in small cancellation quotients of free groups and free products. Paul E. Schupp has obtained similar results for a restricted class of quadratic equations (those without constants). Some of the methods used here were previously employed by Leo P. Comerford and Charles C. Edmunds in their study of algorithmic problems on the solutions of quadratic equations. The principal tools are the construction of cancellation diagrams on compact surfaces with boundary and an analysis of the co-initial graphs of quadratic words.","Let F = and H = be free groups. The generators of F and their inverses are called variables, while the generators of H and their inverses are constants. A word W of F*H is quadratic if each variable which occurs in W occurs exactly twice, with exponent +1 or -1 in each occurrence. The words (alpha)(,1)h(,1)(alpha)(,2)(alpha)(,1)('-1)(alpha)(,2)('-1), (alpha)(,1)('2)(alpha)(,2)('2)(alpha)(,3)('2), and (alpha)(,1)h(,1)(alpha)(,2)h(,2)(alpha)(,1)h(,1)(alpha)(,2) are quadratic. If (alpha)(,1),...,(alpha)(,n) and h(,1),...,h(,k) are the variables and constants, respectively, occurring in W, we write W((alpha)(,1),...,(alpha)(,n); h(,1),...,h(,k)).","Let R be a symmetrized subset of H, and let N be the normal closure of R in H. Let G = H/N and let (phi):H (--->) G be the natural map, so that words of H define elements of G. We look at solutions of W = 1 in G. The tuple (a(,1),...,a(,n)) of words of H is a solution of the equation W((alpha)(,1),...,(alpha)(,n); h(,1),...,h(,k)) in G if the word W(a(,1),...,a(,n); h(,1),...,h(,k)) of H defines the identity element of G.","Much is known about the nature of solutions of quadratic equations in free groups. A free solution of W = 1 in G is induced by a solution of the same equation in H. The solution (a(,1),...,a(,n)) is free if there are words (z(,1),...,z(,n)) of H such that (phi)(z(,i)) = (phi)(a(,i)) for all i and W(z(,1),...,z(,n); h(,1),...,h(,k)) = 1 in H. If we can show that all solutions of W = 1 in a given group G are free, then we can use the classification of solutions in the free group to classify the solutions in G.","If R satisfies sufficiently strong small cancellation conditions, then all solutions of W = 1 in G are free. We first show that a non-free solution induces a reduced R-diagram on a compact surface with boundary. Then we use a combinatorial argument to show that the small cancellation conditions preclude the existence of such a diagram. For this argument we need a lower bound on the Euler characteristics of the surfaces involved; this is where we employ the co-initial graph.","We also obtain similar results when H is a free product. The constants in the equations are arbitrary nontrivial elements of free factors.","Made available in DSpace on 2014-12-14T13:09:50Z (GMT). No. of bitstreams: 1 8203393.pdf: 2178613 bytes, checksum: 0aaa0882908798292814fe2ae396c1ce (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 68362 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","82 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."]},{"key":"dc:title","label":"Title","values":["Solutions of Quadratic Equations in Small Cancellation Groups"]}]}],"canonical_facts":{"dc:creator":["Anderson, Claude Wilson, III"],"dc:date":["2014-12-14T13:09:50Z","10000-01-01","1981"],"dc:description":["The main results are classifications of the solutions of quadratic equations in small cancellation quotients of free groups and free products. Paul E. Schupp has obtained similar results for a restricted class of quadratic equations (those without constants). Some of the methods used here were previously employed by Leo P. Comerford and Charles C. Edmunds in their study of algorithmic problems on the solutions of quadratic equations. The principal tools are the construction of cancellation diagrams on compact surfaces with boundary and an analysis of the co-initial graphs of quadratic words.","Let F = and H = be free groups. The generators of F and their inverses are called variables, while the generators of H and their inverses are constants. A word W of F*H is quadratic if each variable which occurs in W occurs exactly twice, with exponent +1 or -1 in each occurrence. The words (alpha)(,1)h(,1)(alpha)(,2)(alpha)(,1)('-1)(alpha)(,2)('-1), (alpha)(,1)('2)(alpha)(,2)('2)(alpha)(,3)('2), and (alpha)(,1)h(,1)(alpha)(,2)h(,2)(alpha)(,1)h(,1)(alpha)(,2) are quadratic. If (alpha)(,1),...,(alpha)(,n) and h(,1),...,h(,k) are the variables and constants, respectively, occurring in W, we write W((alpha)(,1),...,(alpha)(,n); h(,1),...,h(,k)).","Let R be a symmetrized subset of H, and let N be the normal closure of R in H. Let G = H/N and let (phi):H (--->) G be the natural map, so that words of H define elements of G. We look at solutions of W = 1 in G. The tuple (a(,1),...,a(,n)) of words of H is a solution of the equation W((alpha)(,1),...,(alpha)(,n); h(,1),...,h(,k)) in G if the word W(a(,1),...,a(,n); h(,1),...,h(,k)) of H defines the identity element of G.","Much is known about the nature of solutions of quadratic equations in free groups. A free solution of W = 1 in G is induced by a solution of the same equation in H. The solution (a(,1),...,a(,n)) is free if there are words (z(,1),...,z(,n)) of H such that (phi)(z(,i)) = (phi)(a(,i)) for all i and W(z(,1),...,z(,n); h(,1),...,h(,k)) = 1 in H. If we can show that all solutions of W = 1 in a given group G are free, then we can use the classification of solutions in the free group to classify the solutions in G.","If R satisfies sufficiently strong small cancellation conditions, then all solutions of W = 1 in G are free. We first show that a non-free solution induces a reduced R-diagram on a compact surface with boundary. Then we use a combinatorial argument to show that the small cancellation conditions preclude the existence of such a diagram. For this argument we need a lower bound on the Euler characteristics of the surfaces involved; this is where we employ the co-initial graph.","We also obtain similar results when H is a free product. The constants in the equations are arbitrary nontrivial elements of free factors.","Made available in DSpace on 2014-12-14T13:09:50Z (GMT). No. of bitstreams: 1 8203393.pdf: 2178613 bytes, checksum: 0aaa0882908798292814fe2ae396c1ce (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 68362 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","82 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."],"dc:identifier":["http://hdl.handle.net/2142/68184","(UMI)AAI8203393"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["Solutions of Quadratic Equations in Small Cancellation Groups"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:58Z"}