Algebraic & Geometric Topology

Explicit rank bounds for cyclic covers

Jason DeBlois

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

For a closed, orientable hyperbolic 3–manifold M and an onto homomorphism ϕ: π1(M) that is not induced by a fibration M S1, we bound the ranks of the subgroups ϕ1(n) for n , below, linearly in n. The key new ingredient is the following result: if M is a closed, orientable hyperbolic 3–manifold and S is a connected, two-sided incompressible surface of genus g that is not a fiber or semifiber, then a reduced homotopy in (M,S) has length at most 14g 12.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 3 (2016), 1343-1371.

Dates
Received: 4 November 2013
Revised: 19 October 2015
Accepted: 4 November 2015
First available in Project Euclid: 28 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1511895848

Digital Object Identifier
doi:10.2140/agt.2016.16.1343

Mathematical Reviews number (MathSciNet)
MR3523041

Zentralblatt MATH identifier
1354.57008

Subjects
Primary: 20F05: Generators, relations, and presentations 57M10: Covering spaces
Secondary: 20E06: Free products, free products with amalgamation, Higman-Neumann- Neumann extensions, and generalizations

Keywords
rank rank gradient JSJ decomposition

Citation

DeBlois, Jason. Explicit rank bounds for cyclic covers. Algebr. Geom. Topol. 16 (2016), no. 3, 1343--1371. doi:10.2140/agt.2016.16.1343. https://projecteuclid.org/euclid.agt/1511895848


Export citation

References

  • S Boyer, M Culler, P,B Shalen, X Zhang, Characteristic subsurfaces and Dehn filling, Trans. Amer. Math. Soc. 357 (2005) 2389–2444
  • D Cooper, D,D Long, Virtually Haken Dehn-filling, J. Differential Geom. 52 (1999) 173–187
  • J DeBlois, S Friedl, S Vidussi, Rank gradients of infinite cyclic covers of $3$–manifolds, Michigan Math. J. 63 (2014) 65–81
  • D,B,A Epstein, Curves on $2$–manifolds and isotopies, Acta Math. 115 (1966) 83–107
  • B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton University Press (2012)
  • F Haglund, D,T Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008) 1551–1620
  • A Hatcher, Algebraic topology, Cambridge University Press (2002)
  • J Hempel, $3$–Manifolds, Princeton University Press; University of Tokyo Press (1976)
  • W,H Jaco, P,B Shalen, Seifert fibered spaces in $3$–manifolds, Mem. Amer. Math. Soc. 220, Amer. Math. Soc., Providence, RI (1979)
  • K Johannson, Homotopy equivalences of $3$–manifolds with boundaries, Lecture Notes in Mathematics 761, Springer, Berlin (1979)
  • S Katok, Fuchsian groups, University of Chicago Press (1992)
  • M Lackenby, Expanders, rank and graphs of groups, Israel J. Math. 146 (2005) 357–370
  • T Li, Immersed essential surfaces in hyperbolic $3$–manifolds, Comm. Anal. Geom. 10 (2002) 275–290
  • P Scott, T Wall, Topological methods in group theory, from: “Homological group theory”, (C,T,C Wall, editor), London Math. Soc. Lecture Note Ser. 36, Cambridge University Press (1979) 137–203
  • Z Sela, Acylindrical accessibility for groups, Invent. Math. 129 (1997) 527–565
  • J-P Serre, Trees, corrected reprint of 1st edition, Springer, Berlin (1980)
  • W,P Thurston, A norm for the homology of $3$–manifolds, Mem. Amer. Math. Soc. 339, Amer. Math. Soc., Providence, RI (1986) i–vi and 99–130
  • M,D Tretkoff, A topological approach to the theory of groups acting on trees, J. Pure Appl. Algebra 16 (1980) 323–333
  • G,S Walsh, Incompressible surfaces and spunnormal form, Geom. Dedicata 151 (2011) 221–231
  • R Weidmann, The Nielsen method for groups acting on trees, Proc. London Math. Soc. 85 (2002) 93–118
  • D,T Wise, From riches to raags: $3$–manifolds, right-angled Artin groups, and cubical geometry, CBMS Regional Conference Series in Mathematics 117, Amer. Math. Soc., Providence, RI (2012)
  • D,T Wise, The structure of groups with a quasiconvex hierarchy, preprint (2012) Available at \setbox0\makeatletter\@url http://www.math.mcgill.ca/wise/papers.html {\unhbox0