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Coarse extrinsic geometry: A survey. (English) Zbl 0914.20034
Rivin, Igor (ed.) et al., The Epstein Birthday Schrift dedicated to David Epstein on the occasion of his 60th birthday. Warwick: University of Warwick, Institute of Mathematics, Geom. Topol. Monogr. 1, 341-364 (1998).
The author recalls first the following definition of M. Gromov [contained in Asymptotic invariants of infinite groups, in Geometric group theory II, Lond. Math. Soc. Lect. Note Ser. 182 (1993; Zbl 0841.20039)]. Let \(G\) and \(H\) be finitely generated groups, let \(\Gamma_G\) and \(\Gamma_H\) be the Cayley graphs of \(G\) and \(H\) with respect to some finite sets of generators and let \(i\colon\Gamma_H\to\Gamma_G\) be an embedding. Then, the distorsion function is defined by \[ \text{disto}(R)=\text{Diam}_{\Gamma_H}(\Gamma_H\cap B(R)), \] where \(B(R)\) is the ball of radius \(R\) around the identity of \(\Gamma_G\). If the distorsion function is linear, then \(\Gamma_H\) is said to be quasi-isometrically embedded in \(\Gamma_G\). The author notes that the notion of distorsion makes sense when \(\Gamma_G\) and \(\Gamma_H\) are replaced by arbitrary path-metric spaces, and that this notion is the simplest notion of extrinsic geometry.
In this paper, the author presents a collection of results which concern in particular the distorsion of subgroups in groups, especially in the context of word hyperbolic groups and of the group \(\text{SL}_2(\mathbb{C})\). He also discusses quasi-convexity of subgroups. (In the context of Gromov hyperbolic metric spaces, the notions of quasi-convexity and quasi-isometric embeddings coincide.) The author discusses Gersten’s functional analytic approach to the problem of characterizing quasi-convexity in the context of word hyperbolic groups, as well as other group theoretic approaches to that problem. Another topic in coarse extrinsic geometry which is discussed in this paper is that of boundary extension and of the behaviour at infinity of maps in the context of hyperbolic groups acting on hyperbolic metric spaces. The basic problem discussed is the following: Let \(H\) be a hyperbolic group acting freely and properly discontinuously by isometries on a proper hyperbolic metric space \(X\). Choose a point \(x\in X\) and let \(i\colon\Gamma_H\to X\) be the natural map sending the vertex set of \(\Gamma_H\) to the orbit of \(x\). Then the question is whether \(i\) extends continuously to a map between the hyperbolic boundaries of \(\Gamma_H\) and \(X\). Results related to this question are exposed, in particular the work of Cannon-Thurston in the context of Kleinian groups. The paper contains interesting examples and open problems.
For the entire collection see [Zbl 0901.00063].

20F65 Geometric group theory
57M50 General geometric structures on low-dimensional manifolds
20E07 Subgroup theorems; subgroup growth
20F05 Generators, relations, and presentations of groups
57M07 Topological methods in group theory
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