Relatively hyperbolic groups: geometry and quasi-isometric invariance.

*(English)*Zbl 1175.20032The author studies finitely generated groups which are relatively hyperbolic. (The groups she considers are sometimes called strongly relatively hyperbolic, as opposed to weakly relatively hyperbolic). The groups are assumed to be hyperbolic relative to finitely many proper finitely generated subgroups.

The main result is that the property of relative hyperbolicity is invariant by quasi-isometries. More precisely, she proves the following Theorem: Let \(G\) be a group hyperbolic relative to a family of subgroups \(H_1,\dots,H_n\). If a group \(G'\) is quasi-isometric to \(G\) then \(G'\) is hyperbolic relative to \(H'_1,\dots,H'_m\) where each \(H'_i\) can be embedded quasi-isometrically in \(H_j\) for some \(j=j(i)\in\{1,2,\dots,n\}\).

The question of whether the conclusion of the theorem can be improved to: “\(G'\) is hyperbolic relative to \(H'_1,\dots,H'_m\) where each \(H'_i\) is quasi-isometric to some \(H_j\)” is open.

As a by-product of this work, the author obtains simplified definitions of relative hyperbolicity, in particular, a new definition which is very similar to the one of hyperbolicity, relying on the existence of a central left coset of a peripheral subgroup for every quasi-geodesic triangle.

The main result is that the property of relative hyperbolicity is invariant by quasi-isometries. More precisely, she proves the following Theorem: Let \(G\) be a group hyperbolic relative to a family of subgroups \(H_1,\dots,H_n\). If a group \(G'\) is quasi-isometric to \(G\) then \(G'\) is hyperbolic relative to \(H'_1,\dots,H'_m\) where each \(H'_i\) can be embedded quasi-isometrically in \(H_j\) for some \(j=j(i)\in\{1,2,\dots,n\}\).

The question of whether the conclusion of the theorem can be improved to: “\(G'\) is hyperbolic relative to \(H'_1,\dots,H'_m\) where each \(H'_i\) is quasi-isometric to some \(H_j\)” is open.

As a by-product of this work, the author obtains simplified definitions of relative hyperbolicity, in particular, a new definition which is very similar to the one of hyperbolicity, relying on the existence of a central left coset of a peripheral subgroup for every quasi-geodesic triangle.

Reviewer: Athanase Papadopoulos (Strasbourg)