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

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 of a group , with respect to a generating set , is the distance from to the complement of the radius closed ball, in the word metric defined with respect to . We exhibit a finitely presented group with a finite generating set with respect to which there is no upper bound on the dead-end depth of elements.

**1.**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**0299662**, 10.1090/S0002-9939-1972-0299662-4**2.**Mladen Bestvina.

Questions in geometric group theory.`www.math.utah.edu/bestvina/eprints/questions.dvi`.**3.**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**, 10.1007/BF02671613**4.**Christophe Champetier,*Propriétés statistiques des groupes de présentation finie*, Adv. Math.**116**(1995), no. 2, 197–262 (French, with English summary). MR**1363765**, 10.1006/aima.1995.1067**5.**Sean Cleary and Jennifer Taback,*Combinatorial properties of Thompson’s group 𝐹*, Trans. Amer. Math. Soc.**356**(2004), no. 7, 2825–2849 (electronic). MR**2052598**, 10.1090/S0002-9947-03-03375-0**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****8.**S. Blake Fordham.*Minimal Length Elements of Thompson's group*.

Ph.D. thesis, Brigham Young Univ., 1995.**9.**Walter Parry,*Growth series of some wreath products*, Trans. Amer. Math. Soc.**331**(1992), no. 2, 751–759. MR**1062874**, 10.1090/S0002-9947-1992-1062874-3**10.**Tim R. Riley.

The unbounded depth dead end property is not a group invariant, 2004.`www.math.yale.edu/users/riley`.

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