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...
| Príomhchruthaitheoirí: | , |
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| Rannpháirtithe: | |
| Formáid: | Tráchtas |
| Teanga: | English |
| Foilsithe / Cruthaithe: |
2002
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| Ábhair: |
| Achoimre: | <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> |
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