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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Divergence in lattices in semisimple Lie groups and graphs of groups


Authors: Cornelia Drutu, Shahar Mozes and Mark Sapir
Journal: Trans. Amer. Math. Soc. 362 (2010), 2451-2505
MSC (2000): Primary 20F67; Secondary 20F65
Published electronically: December 16, 2009
MathSciNet review: 2584607
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Abstract: Divergence functions of a metric space estimate the length of a path connecting two points $ A$, $ B$ at distance $ \le n$ avoiding a large enough ball around a third point $ C$. We characterize groups with non-linear divergence functions as groups having cut-points in their asymptotic cones. That property is weaker than the property of having Morse (rank 1) quasi-geodesics. Using our characterization of Morse quasi-geodesics, we give a new proof of the theorem of Farb-Kaimanovich-Masur that states that mapping class groups cannot contain copies of irreducible lattices in semi-simple Lie groups of higher ranks. It also gives a generalization of the result of Birman-Lubotzky-McCarthy about solvable subgroups of mapping class groups not covered by the Tits alternative of Ivanov and McCarthy.

We show that any group acting acylindrically on a simplicial tree or a locally compact hyperbolic graph always has ``many'' periodic Morse quasi-geodesics (i.e. Morse elements), so its divergence functions are never linear. We also show that the same result holds in many cases when the hyperbolic graph satisfies Bowditch's properties that are weaker than local compactness. This gives a new proof of Behrstock's result that every pseudo-Anosov element in a mapping class group is Morse.

On the other hand, we conjecture that lattices in semi-simple Lie groups of higher rank always have linear divergence. We prove it in the case when the $ \mathbb{Q}$-rank is 1 and when the lattice is $ \operatorname{SL}_n(\mathcal{O}_{\mathcal{S}})$ where $ n\ge 3$, $ S$ is a finite set of valuations of a number field $ K$ including all infinite valuations, and $ \mathcal{O}_{\mathcal{S}}$ is the corresponding ring of $ S$-integers.


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Additional Information

Cornelia Drutu
Affiliation: Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, United Kingdom
Email: drutu@maths.ox.ac.uk

Shahar Mozes
Affiliation: Institute of Mathematics, Hebrew University, Jerusalem, Israel

Mark Sapir
Affiliation: Department of Mathematics, SC1522, Vanderbilt University, Nashville, Tennessee 37240
Email: m.sapir@vanderbilt.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04882-X
PII: S 0002-9947(09)04882-X
Received by editor(s): February 7, 2008
Published electronically: December 16, 2009
Additional Notes: The work of the first author was partially supported by the ANR grant GGPG. The research of the second and the third authors was supported in part by a BSF U.S.A.-Israeli grant. In addition, the research of the third author was supported by NSF grants DMS-0455881 and DMS-0700811. All three authors worked on this paper while visiting the Max Planck Institute in Bonn.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.