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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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Limit sets of discrete groups of isometries of exotic hyperbolic spaces
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by Kevin Corlette and Alessandra Iozzi PDF
Trans. Amer. Math. Soc. 351 (1999), 1507-1530 Request permission

Abstract:

Let $\Gamma$ be a geometrically finite discrete group of isometries of hyperbolic space $\mathcal {H}_{\mathbb {F}}^n$, where $\mathbb {F}= \mathbb {R}, \mathbb {C}, \mathbb {H}$ or $\mathbb {O}$ (in which case $n=2$). We prove that the critical exponent of $\Gamma$ equals the Hausdorff dimension of the limit sets $\Lambda (\Gamma )$ and that the smallest eigenvalue of the Laplacian acting on square integrable functions is a quadratic function of either of them (when they are sufficiently large). A generalization of Hopf ergodicity theorem for the geodesic flow with respect to the Bowen-Margulis measure is also proven.
References
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Additional Information
  • Kevin Corlette
  • Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
  • Email: kevin@math.uchicago.edu
  • Alessandra Iozzi
  • Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
  • MR Author ID: 199039
  • Email: iozzi@math.umd.edu
  • Received by editor(s): February 27, 1995
  • Received by editor(s) in revised form: April 15, 1997
  • Additional Notes: K. C. received support from a Sloan Foundation Fellowship, an NSF Presidential Young Investigator award, and NSF grant DMS-9203765. A. I. received support from NSF grants DMS 9001959, DMS 9100383 and DMS 8505550.
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 1507-1530
  • MSC (1991): Primary 58F11; Secondary 53C35, 58F17
  • DOI: https://doi.org/10.1090/S0002-9947-99-02113-3
  • MathSciNet review: 1458321