Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A finitely presented group with unbounded dead-end depth


Authors: Sean Cleary and Tim R. Riley
Journal: Proc. Amer. Math. Soc. 134 (2006), 343-349
MSC (2000): Primary 20F65
DOI: https://doi.org/10.1090/S0002-9939-05-08043-3
Published electronically: August 12, 2005
Erratum: Proc. Amer. Math. Soc. 136 (2008), 2641--2645
MathSciNet review: 2176000
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The dead-end depth of an element $g$ of a group $G$, with respect to a generating set $\mathcal{A}$, is the distance from $g$ to the complement of the radius $d_{\mathcal{A}}(1,g)$ closed ball, in the word metric $d_{\mathcal{A}}$ defined with respect to $\mathcal{A}$. We exhibit a finitely presented group $G$ with a finite generating set with respect to which there is no upper bound on the dead-end depth of elements.


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

  • 1. Gilbert Baumslag.
    A finitely presented metabelian group with a free abelian derived group of infinite rank.
    Proc. Amer. Math. Soc., 35:61-62, 1972. MR 0299662 (45:8710)
  • 2. Mladen Bestvina.
    Questions in geometric group theory. www.math.utah.edu/$\sim$bestvina/eprints/questions.dvi.
  • 3. O. V. Bogopol$'$skii.
    Infinite commensurable hyperbolic groups are bi-Lipschitz equivalent.
    Algebra and Logic, 36(3):155-163, 1997. MR 1485595 (98h:57002)
  • 4. Christophe Champetier.
    Propriétés statistiques des groupes de présentation finie.
    Adv. Math., 116(2):197-262, 1995. MR 1363765 (96m:20056)
  • 5. Sean Cleary and Jennifer Taback.
    Combinatorial properties of Thompson's group $F$.
    Trans. Amer. Math. Soc., 356(7):2825-2849 (electronic), 2004. MR 2052598 (2005b:20074)
  • 6. Sean Cleary and Jennifer Taback.
    Dead end words in lamplighter groups and other wreath products.
    Quarterly Journal of Mathematics, to appear.
  • 7. Pierre de la Harpe.
    Topics in geometric group theory.
    Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. MR 1786869 (2001i:20081)
  • 8. S. Blake Fordham.
    Minimal Length Elements of Thompson's group ${F}$.
    Ph.D. thesis, Brigham Young Univ., 1995.
  • 9. Walter Parry.
    Growth series of some wreath products.
    Trans. Amer. Math. Soc., 331(2):751-759, 1992. MR 1062874 (92h:20061)
  • 10. Tim R. Riley.
    The unbounded depth dead end property is not a group invariant, 2004. www.math.yale.edu/users/riley.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20F65

Retrieve articles in all journals with MSC (2000): 20F65


Additional Information

Sean Cleary
Affiliation: Department of Mathematics, The City College of New York, City University of New York, New York, New York 10031
Email: cleary@sci.ccny.cuny.edu

Tim R. Riley
Affiliation: Department of Mathematics, Yale University, 10 Hillhouse Avenue, P.O. Box 208283, New Haven, Connecticut 06520-8283
Address at time of publication: Department of Mathematics, Cornell University, Ithaca, New York 14853-4201
Email: tim.riley@yale.edu

DOI: https://doi.org/10.1090/S0002-9939-05-08043-3
Keywords: Dead-end depth, lamplighter
Received by editor(s): July 26, 2004
Received by editor(s) in revised form: September 18, 2004
Published electronically: August 12, 2005
Additional Notes: Support for the first author from PSC-CUNY grant #65752 is gratefully acknowledged.
Support for the second author from NSF grant 0404767 is gratefully acknowledged.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society