A finitely presented group with unbounded dead-end depth
HTML articles powered by AMS MathViewer
- by Sean Cleary and Tim R. Riley PDF
- Proc. Amer. Math. Soc. 134 (2006), 343-349 Request permission
Erratum: Proc. Amer. Math. Soc. 136 (2008), 2641-2645.
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
- Gilbert Baumslag, A finitely presented metabelian group with a free abelian derived group of infinite rank, Proc. Amer. Math. Soc. 35 (1972), 61–62. MR 299662, DOI 10.1090/S0002-9939-1972-0299662-4
- Mladen Bestvina. Questions in geometric group theory. www.math.utah.edu/$\sim$bestvina/ eprints/questions.dvi.
- O. V. Bogopol′skiĭ, Infinite commensurable hyperbolic groups are bi-Lipschitz equivalent, Algebra i Logika 36 (1997), no. 3, 259–272, 357 (Russian, with Russian summary); English transl., Algebra and Logic 36 (1997), no. 3, 155–163. MR 1485595, DOI 10.1007/BF02671613
- Christophe Champetier, Propriétés statistiques des groupes de présentation finie, Adv. Math. 116 (1995), no. 2, 197–262 (French, with English summary). MR 1363765, DOI 10.1006/aima.1995.1067
- Sean Cleary and Jennifer Taback, Combinatorial properties of Thompson’s group $F$, Trans. Amer. Math. Soc. 356 (2004), no. 7, 2825–2849. MR 2052598, DOI 10.1090/S0002-9947-03-03375-0
- Sean Cleary and Jennifer Taback. Dead end words in lamplighter groups and other wreath products. Quarterly Journal of Mathematics, to appear.
- Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869
- S. Blake Fordham. Minimal Length Elements of Thompson’s group ${F}$. Ph.D. thesis, Brigham Young Univ., 1995.
- Walter Parry, Growth series of some wreath products, Trans. Amer. Math. Soc. 331 (1992), no. 2, 751–759. MR 1062874, DOI 10.1090/S0002-9947-1992-1062874-3
- Tim R. Riley. The unbounded depth dead end property is not a group invariant, 2004. www.math.yale.edu/users/riley.
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
- MR Author ID: 691109
- Email: tim.riley@yale.edu
- 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
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 343-349
- MSC (2000): Primary 20F65
- DOI: https://doi.org/10.1090/S0002-9939-05-08043-3
- MathSciNet review: 2176000