Tree-graded spaces and asymptotic cones of groups

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 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.

Peripheral fillings of relatively hyperbolic groups

“…Since sides of type (3) were isolated H λ -components of q, the corresponding edges in P are isolated. Hence we may apply Proposition 3.6 to the n-gon P, where the set of components I consists of edges of type (3). Taking into account (21), we obtain…”

“…Relatively hyperbolic groups There are many equivalent definitions of relatively hyperbolic groups [2,3,5,17]. In this paper we use the isoperimetric characterization suggested in [17].…”

“…where [x 3 , u] and [v, x 1 ] are segments of p 3 and p 4 respectively. Using the triangle inequality, it is easy to check that l([v, x 1 ]) ≤ 4δ + 4 and hence l(c) ≤ 8δ + 8.…”

“…Recall that the notion of relative hyperbolicity was introduced in group theory by Gromov in [6] and since then it has been elaborated from different points of view [2,3,5,17].…”

Non-positive curvature and complexity for finitely presented groups

“…[73]. Their geometry and algebraic topology encode a good deal of information about Γ (see [64,83,54]). If Γ is a non-abelian free group, then Cone ω Γ is an everywherebranching R-tree (regardless of the choice of ω).…”