Algebraic & Geometric Topology

A note on spaces of asymptotic dimension one

Koji Fujiwara and Kevin Whyte

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Abstract

Let X be a geodesic metric space with H1(X) uniformly generated. If X has asymptotic dimension one then X is quasi-isometric to an unbounded tree. As a corollary, we show that the asymptotic dimension of the curve graph of a compact, oriented surface with genus g2 and one boundary component is at least two.

Article information

Source
Algebr. Geom. Topol., Volume 7, Number 2 (2007), 1063-1070.

Dates
Received: 30 November 2006
Revised: 1 May 2007
Accepted: 29 May 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796713

Digital Object Identifier
doi:10.2140/agt.2007.7.1063

Mathematical Reviews number (MathSciNet)
MR2336248

Zentralblatt MATH identifier
1141.51010

Subjects
Primary: 51F99: None of the above, but in this section
Secondary: 20F69: Asymptotic properties of groups 54F45: Dimension theory [See also 55M10] 57M50: Geometric structures on low-dimensional manifolds

Keywords
asymptotic dimension quasi-isometry curve graph

Citation

Fujiwara, Koji; Whyte, Kevin. A note on spaces of asymptotic dimension one. Algebr. Geom. Topol. 7 (2007), no. 2, 1063--1070. doi:10.2140/agt.2007.7.1063. https://projecteuclid.org/euclid.agt/1513796713


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References

  • G Baumslag, Wreath products and finitely presented groups, Math. Z. 75 (1960/1961) 22–28
  • G Bell, K Fujiwara, The asymptotic dimension of a curve graph is finite, J. Lond. Math. Soc. (to appear)
  • J Block, S Weinberger, Large scale homology theories and geometry, from: “Geometric topology (Athens, GA, 1993)”, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI (1997) 522–569
  • A N Dranishnikov, Cohomological approach to asymptotic dimension
  • A Dranishnikov, J Smith, Asymptotic dimension of discrete groups, Fund. Math. 189 (2006) 27–34
  • M J Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985) 449–457
  • A Gentimis, Asymptotic dimension of finitely presented group
  • M Gromov, Asymptotic invariants of infinite groups, from: “Geometric group theory, Vol. 2 (Sussex, 1991)”, London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press, Cambridge (1993) 1–295
  • W J Harvey, Boundary structure of the modular group, from: “Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978)”, Ann. of Math. Stud. 97, Princeton Univ. Press, Princeton, N.J. (1981) 245–251
  • T Januszkiewicz, J Swiatkowski, Filling invariants of systolic complexes and groups, Geom. Topol. 11 (2007) 727–758
  • J F Manning, Geometry of pseudocharacters, Geom. Topol. 9 (2005) 1147–1185
  • H A Masur, Y N Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999) 103–149
  • J R Munkres, Topology, second edition, Prentice Hall (2000)
  • S Schleimer, The end of the curve complex
  • J R Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. $(2)$ 88 (1968) 312–334