Title:

Asymptotic invariants of infinite discrete groups

Asymptotic cones. 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 "asymptotic cone" in the limit (this is made precise using techniques of nonstandard analysis). The reward is that in place of the discrete group one has a continuous object "that is amenable to attack by geometric (e.g. topological, infinitesimal) machinery" (to quote Gromov). We give coarse geometric conditions for a metric space X to have Nconnected asymptotic cones. These conditions are expressed in terms of certain filling functions concerning filling Nspheres in an appropriately coarse sense. We interpret the criteria in the case where X 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 Nconnected then Γ is of type F_{N+1} and we provide Nth 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. Combable groups and almostconvex groups. A combing of a finitely generated group Γ is a normal form; that is a choice of word (a combing line) for each group element that satisfies a geometric constraint: nearby group elements have combing lines that fellow travel. An almostconvexity condition concerns the geometry of closed balls in the Cayley graph for Γ. We show that even the most mild combability or almostconvexity restrictions on a finitely presented group already force surprisingly strong constraints on the geometry of its word problem. In both cases we obtain an n! isoperimetric function, and upper bounds of ~ n^{2} on both the minimal isodiametric function and the filling length function.
