On the asymptotic geometry of abelian-by-cyclic groups

By Benson Farb, Lee Mosher. 2000

91 citations recorded by the source corpus.

Gromov’s Polynomial Growth Theorem [Gro81] states that the property of having polynomial growth characterizes virtually nilpotent groups among all finitely generated groups. Gromov’s theorem inspired the more general problem (see, e.g. [GdlH91]) of understanding to what extent the asymptotic geometry of a finitelygenerated solvable group determines its algebraic structure—in short, are solvable groups quasi-isometrically rigid? In general they aren’t: very recently A. Dioubina [Dio99] has found a solvable group which is quasi-isometric to a group which is not virtually solvable; these groups are finitely generated but not finitely presentable. In the opposite direction, first steps in identifying quasi-isometrically rigid solvable groups which are not virtually nilpotent were taken for a special class of examples, the solvable BaumslagSolitar groups, in [FM98] and [FM99b]. The goal of the present paper is to show that a much broader class of solvable groups, the class of finitely-presented, nonpolycyclic, abelian-bycyclic groups, is characterized among all finitely-generated groups by its quasi-isometry type. We also give a complete quasi-isometry classification of the groups in this class; such a classification for nilpotent groups remains a major open question. Motivated by these results, we offer a conjectural picture of quasi-isometric classification and rigidity for polycyclic abelianby-cyclic groups in §10.1.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7e8944f49c3deab20ac3ea9a7ab12d4355dce183

How researchers see this paper

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In Applied Mathematics

1. K. Whyte
2. D. Fisher
3. A. Eskin
4. Michah Sageev
5. Lee Mosher
6. Ilya Kapovich
7. Nadia Benakli
8. Bruce Kleiner

In Geometry and Topology

1. K. Whyte
2. D. Fisher
3. A. Eskin
4. Michah Sageev
5. Lee Mosher
6. Ilya Kapovich
7. Nadia Benakli
8. Bruce Kleiner

In Mathematical Physics

1. K. Whyte
2. Michah Sageev
3. D. Fisher
4. A. Eskin
5. Ilya Kapovich
6. Lee Mosher
7. Bruce Kleiner
8. Denis V. Osin

Citations in the corpus

• Quasi-isometries and rigidity of solvable groups (2005, methodology): “See particularly the work of Farb and Mosher on the solvable Baumslag-Solitar groups [FM1, FM2] as well as later work of Farb-Mosher, Mosher-Sageev-Whyte and Wortman [ FM3 , MSW, W]. The methods used in all of these works depend essentially on topological arguments based on the explicit structure of”
• Boundaries of hyperbolic groups (2002, background): “Rigidity ( see [164], [127] , [78], [81]) and the renewed prominence of CAT (k) (or Cartan-Alexandrov-Toponogov) Geometry (see [5], [45]).”
• Quasi-actions on trees I. Bounded valence (2000, methodology): “The proof follows [FM00, §7.2, Step 2] very closely; here is a sketch.”
• Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric to Cayley graphs (2006, methodology): “This then follows immediately from Theorem 2.1 and [ FM3 , Theorem 5.1].”