Second Topic: Quasi-isometries and Splittings of Groups
نویسنده
چکیده
The word metric on a group is independent of the choice of generating set, so the quasiisometry type of a group is well defined. We say that two groups are (abstractly) commensurable if they are related to one another by any (finite) sequence of finite processes: taking finite-index subgroups or supergroups, finite quotients or extensions. Groups which are commensurable are quasi-isometric. The property of having polynomial growth is invariant under quasi-isometry, so Gromov’s Theorem can be viewed as saying that the class of nilpotent groups is quasi-isometrically rigid; i.e., any group which is quasi-isometric to a group in the class is commensurable with a group in the class. Similarly, Stallings’ Theorem on ends shows that the class of groups which split over finite groups is quasi-isometrically rigid. These results have motivated the program, introduced by Gromov [G83, G93], of classifying all finitely generated groups up to quasi-isometry. Gromov’s theorem for nilpotent
منابع مشابه
Group Splittings and Asymptotic Topology
It is a consequence of the theorem of Stallings on groups with many ends that splittings over finite groups are preserved by quasi-isometries. In this paper we use asymptotic topology to show that group splittings are preserved by quasi-isometries in many cases. Roughly speaking we show that splittings are preserved under quasi-isometries when the vertex groups are fundamental groups of aspheri...
متن کاملar X iv : m at h / 02 01 31 2 v 1 [ m at h . G R ] 3 1 Ja n 20 02 GROUP SPLITTINGS AND ASYMPTOTIC TOPOLOGY
It is a consequence of the theorem of Stallings on groups with many ends that splittings over finite groups are preserved by quasi-isometries. In this paper we use asymptotic topology to show that group splittings are preserved by quasi-isometries in many cases. Roughly speaking we show that splittings are preserved under quasi-isometries when the vertex groups are fundamental groups of aspheri...
متن کاملA Note on a Theorem of Bowditch
Bowditch showed that a one-ended hyperbolic group which is not a triangle group splits over a two-ended group if and only if its boundary has a local cut point. As a corollary one obtains that splittings of hyperbolic groups over twoended groups are preserved under quasi-isometries. In this note we give a more direct proof of this corollary.
متن کاملRigidity of quasi-isometries for symmetric spaces and Euclidean buildings
for all x ∈ X . Quasi-isometries occur naturally in the study of the geometry of discrete groups since the length spaces on which a given finitely generated group acts cocompactly and properly discontinuously by isometries are quasi-isometric to one another [Gro]. Quasi-isometries also play a crucial role in Mostow’s proof of his rigidity theorem: the theorem is proved by showing that equivaria...
متن کاملQuasi-isometries and Rigidity of Solvable Groups
In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasiisometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to R⋉Rn where the semidirect product is defined by a diagonalizable matrix of determinant one with ...
متن کامل