Tree-graded spaces and asymptotic cones of groups

By Cornelia Drutu, Mark V. Sapir. 2004

416 citations recorded by the source corpus.

Abstract We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic groups, and to construct the first example of a finitely generated group with a continuum of non- π 1 -equivalent asymptotic cones. Note that by a result of Kramer, Shelah, Tent and Thomas, continuum is the maximal possible number of different asymptotic cones of a finitely generated group, provided that the Continuum Hypothesis is true.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/117b954a2fe29760813c405db2b6fc6051514857

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Citations in the corpus

Constructing group actions on quasi-trees and applications to mapping class groups (2010, background): “s. In fact they prove more including some information on the geometry of the image of the embedding, but their theorem does not imply Theorem C. They use the notion of tree-graded space introduced in [DrS]. We now make a few comments on verifying the axioms (P0)-(P2) in this setting, as this is the ”
Contracting boundaries of CAT(0) spaces (2013, background): “These notions have proved fruitful in analyzing right-angled Artin groups (RAAGs) [3], Teichmüller space [2, 7–9], the mapping class group [2], CAT(0) spaces [4, 5, 30], and Out(Fn) [1] among others (see also [13, 14, 21, 22]).”
Statistically convex-cocompact actions of groups with contracting elements (2016, background): “perly and co-compactly on a -hyperbolic space (Y;d) ž. See [48], [44]. ● RelHyp =Ž a relatively hyperbolic group Gacts on a Cayley graph (Y;d) with respect to a generating set Sž. See [16], [68] and [30]. ● Floyd = Ž a group Gwith non-trivial Floyd boundary acts on a Cayley graph (Y;d) with respect ”
On strongly quasiconvex subgroups (2017, background): “on-hyperbolic peripheral subgroups of a relatively hyperbolic group and non-hyperbolic hyperbolically embedded subgroups of a ﬁnitely generated group are non-stable strongly quasiconvex subgroups(see [DS05] and [Sis16]). In Section 7, we also provide many examples of non-stable strongly quasiconvex ”
Acylindrical actions on projection complexes (2017, background): “ith the same endpoint. Up to replacing αwith a path whose image is contained in α, we can assume that αis injective. The conclusion that γis contained in a uniform neighborhood 1In the terminology of [DS05], C K(Y) is asymptotically tree-graded with respect to Y. ACYLINDRICAL ACTIONS ON PROJECTION C”
On metric relative hyperbolicity (2012, methodology): “subsets, is said to satisfy (RH0) if Psatises ( 1) and ( 2) and Xis weakly () asymptotically tree-graded with respect to P. We record the following result that will often be tacitly used. Lemma 2.6. [DS05a] Let P;Qbe collections of subsets of the geodesic metric space X. Suppose that there exists K0”
The SQ-universality and residual properties of relatively hyperbolic groups (2006, background): “The notion of a group hyperbolic relative to a collection of subgroups was originally suggested by Gromov [9] and since then it has been elaborated from different points of view [3, 6, 5, 21].”
Geometry and rigidity of mapping class groups (2008, background): “4 to give a characterization of the tree-graded structure (à la Druţu and Sapir [12]) of the cone of MCG.”
Some geometric groups with Rapid Decay (2003, background): “12 (3) of [11], quasi-geodesics in the Cayley graph of G are at bounded Hausdorff distance from any geodesic in the coned-off graph Γ̂.”