Quasi-isometries and rigidity of solvable groups

By A. Eskin, D. Fisher, K. Whyte. 2005

94 citations recorded by the source corpus.

In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to $R \ltimes R^n$ where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasi-isometries of the solvable Lie group. Our classification of self quasi-isometries for $R \ltimes \R^n$ proves a conjecture made by Farb and Mosher in [FM4]. Our techniques for studying quasi-isometries extend to some other classes of groups and spaces. In particular, we characterize groups quasi-isometric to any lamplighter group, answering a question of de la Harpe [dlH]. Also, we prove that certain Diestel-Leader graphs are not quasi-isometric to any finitely generated group, verifying a conjecture of Diestel and Leader from [DL] and answering a question of Woess from [SW],[Wo1]. We also prove that certain non-unimodular, non-hyperbolic solvable Lie groups are not quasi-isometric to finitely generated groups. The results in this paper are contributions to Gromov's program for classifying finitely generated groups up to quasi-isometry [Gr2]. We introduce a new technique for studying quasi-isometries, which we refer to as "coarse differentiation".

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3dd946cd51f40b413677f975eedf24e1a8cee1b7

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Citations in the corpus

Metric differentiation, monotonicity and maps to L1 (2009, methodology): “We note that Lee-Raghavendra [27] have used a similar combination of ideas in the context of finite graphs: they use a form of metric differentiation— the coarse differentiation of Eskin-Fisher-Whyte [ 16 ]—together with essentially the same notion of monotonicity as above.”
Quasi-isometric classification of graph manifold groups (2006, background): “W e also classify the quasi-isom etry types offundam entalgroups ofgraph-m anifolds with boundary in term s ofcertain nite two-colored graphs.”
Horocyclic products of trees (2006, background): “... for Diestel and Leader, who were the first [16] to introduce the graph DL(2,3) in attempt to answer a question raised by Woess [33, 31]: “is there a locally finite vertex-transitive graph which is not quasiisometric with a Cayley graph of some finitely generated group ?” (Recall that a graph is ”
Geometry and rigidity of mapping class groups (2008, background): “• Graphs of n-dimensional Poincaré duality groups, for some fixed n (Farb-Mosher, Whyte, Mosher-Sageev-Whyte, Papasoglu [18, 19, 45, 46, 49, 56]).”