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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Divergence in lattices in semisimple Lie groups and graphs of groups
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by Cornelia Druţu, Shahar Mozes and Mark Sapir PDF
Trans. Amer. Math. Soc. 362 (2010), 2451-2505 Request permission

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.

References
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Additional Information
  • Cornelia Druţu
  • 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
  • MR Author ID: 264125
  • Mark Sapir
  • Affiliation: Department of Mathematics, SC1522, Vanderbilt University, Nashville, Tennessee 37240
  • MR Author ID: 189574
  • Email: m.sapir@vanderbilt.edu
  • 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.
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 362 (2010), 2451-2505
  • MSC (2000): Primary 20F67; Secondary 20F65
  • DOI: https://doi.org/10.1090/S0002-9947-09-04882-X
  • MathSciNet review: 2584607