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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Quasiisometric embedding of the fundamental group of an orthogonal graph-manifold into a product of metric trees

Author: A. Smirnov
Translated by: the author
Original publication: Algebra i Analiz, tom 24 (2012), nomer 5.
Journal: St. Petersburg Math. J. 24 (2013), 811-821
MSC (2010): Primary 57M15
Published electronically: July 24, 2013
MathSciNet review: 3087825
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Abstract | References | Similar Articles | Additional Information

Abstract: In every dimension $ n\ge 3$, a class of orthogonal graph-manifolds is introduced. It is proved that the fundamental group of any orthogonal graph-manifold embeds quasiisometrically into a product of $ n$ trees. As a consequence, it is shown that the asymptotic dimension and the linearly-controlled asymptotic dimensions of such a group are equal to $ n$.

References [Enhancements On Off] (What's this?)

  • 1. J. A. Behrstock and W. D. Neumann, Quasi-isometric classification of graph manifold groups, Duke Math. J. 141 (2008), no. 2, 217-240. MR 2376814 (2009c:20070)
  • 2. D. Yu. Burago, Yu. D. Burago, and S. V. Ivanov, A course in metric geometry, Inst. Komp'yuter. Issled., Moscow-Izhevsk, 2005; English variant, Grad. Stud. Math., vol. 33, Amer. Math. Soc., Providence, RI, 2001. MR 1835418 (2002e:53053)
  • 3. G. Bell and A. Dranishnikov, On Asymptotic Dimension of Groups Acting on Trees, Geometriae Dedicata, 103 (2004), no. 1, 89-101. MR 2034954 (2005b:20078)
  • 4. S. V. Buyalo and V. L. Kobel'skiĭ, Generalized graphmanifolds of nonpositive curvature, Algebra i Analiz 11 (1999), no. 2, 64-87; English transl., St. Petersburg Math. J. 11 (2000), no. 2, 251-268. MR 1702579 (2001f:53062)
  • 5. S. V. Buyalo and V. Schroeder, Elements of asymptotic geometry, EMS Monogr. in Math., Europ. Math. Soc., Zurich, 2007. MR 2327160 (2009a:53068)
  • 6. M. Gromov., Asymptotic invariants of infinite groups, Geometric Group Theory, Vol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1-295. MR 1253544 (95m:20041)
  • 7. D. Hume and A. Sisto, Embedding universal covers of graph manifolds in products of trees, Preprint arXiv:math.GT/1112.0263.
  • 8. M. Kapovich and B. Leeb, 3-manifold groups and nonpositive curvature, Geom. Funct. Anal. 8 (1998), 841-852. MR 1650098 (2000a:57040)
  • 9. J. Roe, Lectures on coarse geometry, Univ. Lecture Ser., vol. 31, Amer. Math. Soc., Providence, RI, 2003. MR 2007488 (2004g:53050)

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Additional Information

A. Smirnov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

Keywords: Graph-manifold, quasiisometric embedding
Received by editor(s): December 14, 2011
Published electronically: July 24, 2013
Additional Notes: Supported by RFBR (grant no. 11-01-00302-a)
Article copyright: © Copyright 2013 American Mathematical Society

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