Asymptotic invariants of infinite discrete groups
<p><b>Asymptotic cones.</b> A finitely generated group has a word metric, which one can scale and thereby view the group from increasingly distant vantage points. The group coalesces to an "<em>asymptotic cone"</em> in the limit (this is made precise using tec...
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| Format: | Thesis |
| Language: | English |
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2002
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| _version_ | 1826316214613311488 |
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| author | Riley, T Timothy Riley |
| author2 | Bridson, M |
| author_facet | Bridson, M Riley, T Timothy Riley |
| author_sort | Riley, T |
| collection | OXFORD |
| description | <p><b>Asymptotic cones.</b> A finitely generated group has a word metric, which one can scale and thereby view the group from increasingly distant vantage points. The group coalesces to an "<em>asymptotic cone"</em> in the limit (this is made precise using techniques of non-standard analysis). The reward is that in place of the discrete group one has a continuous object "that is amenable to attack by geometric (<em>e.g.</em> topological, infinitesimal) machinery" (to quote Gromov).</p> <p>We give coarse geometric conditions for a metric space X to have <em>N</em>-connected asymptotic cones. These conditions are expressed in terms of certain filling functions concerning filling <em>N</em>-spheres in an appropriately coarse sense.</p> <p>We interpret the criteria in the case where <em>X</em> is a finitely generated group Γ with a word metric. This leads to upper bounds on filling functions for groups with simply connected cones -- in particular they have linearly bounded filling length functions. We prove that if all the asymptotic cones of Γ are <em>N</em>-connected then Γ is of type F<sub><em>N</em>+1</sub> and we provide <em>N</em>-th order isoperimetric and isodiametric functions. Also we show that the asymptotic cones of a virtually polycyclic group Γ are all contractible if and only if Γ is virtually nilpotent.</p> <p><b>Combable groups and almost-convex groups.</b> A <em>combing</em> of a finitely generated group Γ is a normal form; that is a choice of word (a <em>combing line</em>) for each group element that satisfies a geometric constraint: nearby group elements have combing lines that <em>fellow travel</em>. An <em>almost-convexity condition</em> concerns the geometry of closed balls in the Cayley graph for Γ.</p> <p>We show that even the most mild <em>combability</em> or <em>almost-convexity</em> restrictions on a finitely presented group already force surprisingly strong constraints on the geometry of its word problem. In both cases we obtain an <em>n</em>! isoperimetric function, and upper bounds of ~ <em>n</em><sup>2</sup> on both the minimal isodiametric function and the filling length function.</p> |
| first_indexed | 2024-03-06T20:30:45Z |
| format | Thesis |
| id | oxford-uuid:30f42f4c-e592-44c2-9954-7d9e8c1f3d13 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-12-09T03:40:49Z |
| publishDate | 2002 |
| record_format | dspace |
| spelling | oxford-uuid:30f42f4c-e592-44c2-9954-7d9e8c1f3d132024-12-07T11:48:30ZAsymptotic invariants of infinite discrete groupsThesishttp://purl.org/coar/resource_type/c_db06uuid:30f42f4c-e592-44c2-9954-7d9e8c1f3d13Group theory and generalizations (mathematics)EnglishOxford University Research Archive - Valet2002Riley, TTimothy RileyBridson, M<p><b>Asymptotic cones.</b> A finitely generated group has a word metric, which one can scale and thereby view the group from increasingly distant vantage points. The group coalesces to an "<em>asymptotic cone"</em> in the limit (this is made precise using techniques of non-standard analysis). The reward is that in place of the discrete group one has a continuous object "that is amenable to attack by geometric (<em>e.g.</em> topological, infinitesimal) machinery" (to quote Gromov).</p> <p>We give coarse geometric conditions for a metric space X to have <em>N</em>-connected asymptotic cones. These conditions are expressed in terms of certain filling functions concerning filling <em>N</em>-spheres in an appropriately coarse sense.</p> <p>We interpret the criteria in the case where <em>X</em> is a finitely generated group Γ with a word metric. This leads to upper bounds on filling functions for groups with simply connected cones -- in particular they have linearly bounded filling length functions. We prove that if all the asymptotic cones of Γ are <em>N</em>-connected then Γ is of type F<sub><em>N</em>+1</sub> and we provide <em>N</em>-th order isoperimetric and isodiametric functions. Also we show that the asymptotic cones of a virtually polycyclic group Γ are all contractible if and only if Γ is virtually nilpotent.</p> <p><b>Combable groups and almost-convex groups.</b> A <em>combing</em> of a finitely generated group Γ is a normal form; that is a choice of word (a <em>combing line</em>) for each group element that satisfies a geometric constraint: nearby group elements have combing lines that <em>fellow travel</em>. An <em>almost-convexity condition</em> concerns the geometry of closed balls in the Cayley graph for Γ.</p> <p>We show that even the most mild <em>combability</em> or <em>almost-convexity</em> restrictions on a finitely presented group already force surprisingly strong constraints on the geometry of its word problem. In both cases we obtain an <em>n</em>! isoperimetric function, and upper bounds of ~ <em>n</em><sup>2</sup> on both the minimal isodiametric function and the filling length function.</p> |
| spellingShingle | Group theory and generalizations (mathematics) Riley, T Timothy Riley Asymptotic invariants of infinite discrete groups |
| title | Asymptotic invariants of infinite discrete groups |
| title_full | Asymptotic invariants of infinite discrete groups |
| title_fullStr | Asymptotic invariants of infinite discrete groups |
| title_full_unstemmed | Asymptotic invariants of infinite discrete groups |
| title_short | Asymptotic invariants of infinite discrete groups |
| title_sort | asymptotic invariants of infinite discrete groups |
| topic | Group theory and generalizations (mathematics) |
| work_keys_str_mv | AT rileyt asymptoticinvariantsofinfinitediscretegroups AT timothyriley asymptoticinvariantsofinfinitediscretegroups |