Mathematics – Group Theory
Scientific paper
2000-08-29
Mathematics
Group Theory
48 pages
Scientific paper
We show that every word hyperbolic, surface-by-(noncyclic) free group Gamma is as rigid as possible: the quasi-isometry group of Gamma equals the abstract commensurator group Comm(Gamma), which in turn contains Gamma as a finite index subgroup. As a corollary, two such groups are quasi-isometric if and only if they are commensurable, and any finitely generated group quasi-isometric to Gamma must be weakly commensurable with Gamma. We use quasi-isometries to compute Comm(Gamma) explicitly, an example of how quasi-isometries can actually detect finite index information. The proofs of these theorems involve ideas from coarse topology, Teichmuller geometry, pseudo-Anosov dynamics, and singular solv-geometry.
Farb Benson
Mosher Lee
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