{"id":{"repo_id":"birmingham","oai_identifier":"oai:etheses.bham.ac.uk:1626"},"canonical_url":"https://search.dev.ndltd.org/etd/birmingham/oai:etheses.bham.ac.uk:1626","repository":{"repo_id":"birmingham","name":"University of Birmingham","base_url":"https://etheses.bham.ac.uk/cgi/oai2"},"display":{"title":"A family of biaffine geometries and their resulting amalgams","abstract":"Let \\(\\Pi\\) be a thick polar space of rank \\(n\\) at least three. Pick a hyperplane \\(F\\) of \\(\\Pi\\) and \\(H\\) of \\(\\Pi\\)\\(^{\\ast}\\). Define the elements of a biaffine polar space \\(\\Gamma\\) to be those elements of \\(\\Pi\\) which are not contained in \\(F\\), or dually in \\(H\\). We show that \\(\\Gamma\\) is a non empty geometry which is simply connected, except for a few small exceptions for \\(\\Pi\\). We give two pairs of examples with ag-transitive groups, which lead to amalgam results for recognising either one of \\(q\\)\\(^6\\) : \\(SU\\)\\(_3\\)\\((q)\\) or \\(G\\)\\(_2\\)\\((q)\\), or one of \\(q\\)\\(^7\\) : \\(G\\)\\(_2\\)\\((q)\\) or \\(Spin\\)\\(^7\\)\\((q)\\). Also, we give details of a computer program to calculate the fundamental group of a given geometry.","abstract_html":"Let \\(\\Pi\\) be a thick polar space of rank \\(n\\) at least three. Pick a hyperplane \\(F\\) of \\(\\Pi\\) and \\(H\\) of \\(\\Pi\\)<span class=\"etd-inline-math\"><sup>\\ast</sup></span>. Define the elements of a biaffine polar space \\(\\Gamma\\) to be those elements of \\(\\Pi\\) which are not contained in \\(F\\), or dually in \\(H\\). We show that \\(\\Gamma\\) is a non empty geometry which is simply connected, except for a few small exceptions for \\(\\Pi\\). We give two pairs of examples with ag-transitive groups, which lead to amalgam results for recognising either one of \\(q\\)<span class=\"etd-inline-math\"><sup>6</sup></span> : \\(SU\\)<span class=\"etd-inline-math\"><sub>3</sub></span>\\((q)\\) or \\(G\\)<span class=\"etd-inline-math\"><sub>2</sub></span>\\((q)\\), or one of \\(q\\)<span class=\"etd-inline-math\"><sup>7</sup></span> : \\(G\\)<span class=\"etd-inline-math\"><sub>2</sub></span>\\((q)\\) or \\(Spin\\)<span class=\"etd-inline-math\"><sup>7</sup></span>\\((q)\\). Also, we give details of a computer program to calculate the fundamental group of a given geometry.","abstract_has_math":true,"creators":["McInroy, Justin Fergus"],"institution":"University of Birmingham","degree_name":"d_ph","degree_level":"d_ph","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-07","date_published":"2011-07","updated_at":"2026-07-24T01:11:37Z","subjects":["QA Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.sponsor","label":"Sponsor","values":["epsrc"]},{"key":"dc:creator","label":"Author","values":["McInroy, Justin Fergus"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-07"]},{"key":"dc:date.issued","label":"Date","values":["2011-07"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["College of Engineering & Physical Sciences","School of Mathematics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Birmingham"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["http://etheses.bham.ac.uk//id/eprint/1626/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["d_ph"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["d_ph"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["QA Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://etheses.bham.ac.uk//id/eprint/1626/1/McInroy11PhD.pdf","http://etheses.bham.ac.uk//id/eprint/1626/2/Decl_IS_McInroy11PhD.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let \\(\\Pi\\) be a thick polar space of rank \\(n\\) at least three. Pick a hyperplane \\(F\\) of \\(\\Pi\\) and \\(H\\) of \\(\\Pi\\)\\(^{\\ast}\\). Define the elements of a biaffine polar space \\(\\Gamma\\) to be those elements of \\(\\Pi\\) which are not contained in \\(F\\), or dually in \\(H\\). We show that \\(\\Gamma\\) is a non empty geometry which is simply connected, except for a few small exceptions for \\(\\Pi\\). We give two pairs of examples with ag-transitive groups, which lead to amalgam results for recognising either one of \\(q\\)\\(^6\\) : \\(SU\\)\\(_3\\)\\((q)\\) or \\(G\\)\\(_2\\)\\((q)\\), or one of \\(q\\)\\(^7\\) : \\(G\\)\\(_2\\)\\((q)\\) or \\(Spin\\)\\(^7\\)\\((q)\\). Also, we give details of a computer program to calculate the fundamental group of a given geometry."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["A family of biaffine geometries and their resulting amalgams"]}]}],"canonical_facts":{"dc:contributor.sponsor":["epsrc"],"dc:creator":["McInroy, Justin Fergus"],"dc:date":["2011-07"],"dc:date.issued":["2011-07"],"dc:description.abstract":["Let \\(\\Pi\\) be a thick polar space of rank \\(n\\) at least three. Pick a hyperplane \\(F\\) of \\(\\Pi\\) and \\(H\\) of \\(\\Pi\\)\\(^{\\ast}\\). Define the elements of a biaffine polar space \\(\\Gamma\\) to be those elements of \\(\\Pi\\) which are not contained in \\(F\\), or dually in \\(H\\). We show that \\(\\Gamma\\) is a non empty geometry which is simply connected, except for a few small exceptions for \\(\\Pi\\). We give two pairs of examples with ag-transitive groups, which lead to amalgam results for recognising either one of \\(q\\)\\(^6\\) : \\(SU\\)\\(_3\\)\\((q)\\) or \\(G\\)\\(_2\\)\\((q)\\), or one of \\(q\\)\\(^7\\) : \\(G\\)\\(_2\\)\\((q)\\) or \\(Spin\\)\\(^7\\)\\((q)\\). Also, we give details of a computer program to calculate the fundamental group of a given geometry."],"dc:format":["application/pdf"],"dc:identifier.uri":["http://etheses.bham.ac.uk//id/eprint/1626/1/McInroy11PhD.pdf","http://etheses.bham.ac.uk//id/eprint/1626/2/Decl_IS_McInroy11PhD.pdf"],"dc:publisher.department":["College of Engineering & Physical Sciences","School of Mathematics"],"dc:publisher.institution":["University of Birmingham"],"dc:relation.isreferencedby":["http://etheses.bham.ac.uk//id/eprint/1626/"],"dc:subject":["QA Mathematics"],"dc:title":["A family of biaffine geometries and their resulting amalgams"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["d_ph"],"dc:type.qualificationname":["d_ph"]},"updated_at":"2026-07-24T01:11:37Z"}