Higher connectedness of asymptotic cones

Riley, Tim

doi:10.1016/s0040-9383(03)00002-8

Cited by 27 publications

(35 citation statements)

References 28 publications

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“…The evolution of a color 21 Note that Stallings' group was the first example of a finitely presented group which is not of type F P 3 . Thus by [71] and a previous results of Riley [203] and Papasoglu [196] this group is the first example of a finitely presented group with simply connected but not 2-connected asymptotic cones. 22 The R. Thompson group F consists of all piecewise-linear increasing bijective functions [0, 1] → [0, 1] whose derivatives have finitely many dyadic break points and the slopes of linear pieces are powers of 2.…”

Section: 1b Nilpotent Groups From Partsupporting

confidence: 58%

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Asymptotic invariants, complexity of groups and related problems

Sapir

2011

Bull. Math. Sci.

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Section: 1b Nilpotent Groups From Partsupporting

confidence: 58%

“…Riley [203] proved that under the assumptions of Theorem 2.27, the group also has linear FL function (and hence linear FFFL function as well).…”

Section: Asymptotic Cones and Dehn Functionsmentioning

confidence: 99%

Asymptotic invariants, complexity of groups and related problems

Sapir

2011

Bull. Math. Sci.

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“…These techniques streamlined parts of the proof, replacing an iterated procedure of passing to subsequences with one ultralimit, and broadened the applicability of Gromov's construction, Gromov's original definition of asymptotic cone only applied to nilpotent groups whereas the ultrafied definition works for any finitely generated group. (For other applications of asymptotic cones in geometric group theory see: Burillo [17], Bridson [12], Druţu [18], Druţu and Sapir [19], Kapovich and Leeb [36] Kleiner and Leeb [38], Kramer, Shelah, Tent and Thomas [39] and Riley [50]. )…”

Section: Asymptotic Conesmentioning

confidence: 99%

Asymptotic geometry of the mapping class group and Teichmüller space

Behrstock¹

2006

Geom. Topol.

142

214

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In this work, we study the asymptotic geometry of the mapping class group and Teichmuller space. We introduce tools for analyzing the geometry of "projection" maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. We deduce several applications of this analysis. One of which is that the asymptotic cone of the mapping class group of any surface is tree-graded in the sense of Druţu and Sapir; this treegrading has several consequences including answering a question of Druţu and Sapir concerning relatively hyperbolic groups. Another application is a generalization of the result of Brock and Farb that for low complexity surfaces Teichmüller space, with the Weil-Petersson metric, is ı -hyperbolic. Although for higher complexity surfaces these spaces are not ı -hyperbolic, we establish the presence of previously unknown negative curvature phenomena in the mapping class group and Teichmüller space for arbitrary surfaces.20F67; 30F60, 57M07, 20F65

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“…If Conjecture 4.3 holds, then by Theorem 1.3 the linear bound on the gallery length would also recapture the linear bound on filling length of [13,20].…”

Section: Corollary 66 Assume Conjecture 43 Holds If P Is a Finitementioning

confidence: 94%

Some Duality Conjectures for Finite Graphs and Their Group Theoretic Consequences

Gersten¹,

Riley²

2005

Proceedings of the Edinburgh Mathematical Society

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We pose some graph theoretic conjectures about duality and the diameter of maximal trees in planar graphs, and we give innovations in the following two topics in geometric group theory, where the conjectures have applications.Central extensions. We describe an electrostatic model concerning how van Kampen diagrams change when one takes a central extension of a group. Modulo the conjectures, this leads to a new proof that finitely generated class c nilpotent groups admit degree c + 1 polynomial isoperimetric functions.Filling functions. We collate and extend results about interrelationships between filling functions for finite presentations of groups. We use the electrostatic model in proving that the gallery length filling function, which measures the diameter of the duals of diagrams, is qualitatively the same as a filling function DlogA, concerning the sum of the diameter with the logarithm of the area of a diagram. We show that the conjectures imply that the space-complexity filling function filling length essentially equates to gallery length. We give linear upper bounds on these functions for a number of classes of groups including fundamental groups of compact geometrizable 3-manifolds, certain graphs of groups, and almost convex groups. Also we define restricted filling functions which concern diagrams with uniformly bounded vertex valence, and we show that, assuming the conjectures, they reduce to just two filling functions-the analogues of non-deterministic space and time.

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Higher connectedness of asymptotic cones

Cited by 27 publications

References 28 publications

Asymptotic invariants, complexity of groups and related problems

Asymptotic invariants, complexity of groups and related problems

Asymptotic geometry of the mapping class group and Teichmüller space

Some Duality Conjectures for Finite Graphs and Their Group Theoretic Consequences

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