Duke Mathematical Journal

Simple closed curves, finite covers of surfaces, and power subgroups of Out(Fn)

Justin Malestein and Andrew Putman

Full-text: Access denied (no subscription detected)

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

We construct examples of finite covers of punctured surfaces where the first rational homology is not spanned by lifts of simple closed curves. More generally, for any set OFn which is contained in the union of finitely many Aut(Fn)-orbits, we construct finite-index normal subgroups of Fn whose first rational homology is not spanned by powers of elements of O. These examples answer questions of Farb and Hensel, Kent, Looijenga, and Marché. We also show that the quotient of Out(Fn) by the subgroup generated by kth powers of transvections often contains infinite-order elements, strengthening a result of Bridson and Vogtmann that it is often infinite. Finally, for any set OFn which is contained in the union of finitely many Aut(Fn)-orbits, we construct integral linear representations of free groups that have infinite image and that map all elements of O to torsion elements.

Article information

Source
Duke Math. J., Volume 168, Number 14 (2019), 2701-2726.

Dates
Received: 25 March 2018
Revised: 12 January 2019
First available in Project Euclid: 10 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1568081121

Digital Object Identifier
doi:10.1215/00127094-2019-0022

Subjects
Primary: 57M10: Covering spaces
Secondary: 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx] 20C05: Group rings of finite groups and their modules [See also 16S34]

Keywords
covering spaces surfaces restricted Lie algebras central series p-groups transvections linear representations mapping class group outer automorphism group of the free group representations of finite groups simple closed curves primitives

Citation

Malestein, Justin; Putman, Andrew. Simple closed curves, finite covers of surfaces, and power subgroups of $\operatorname{Out}(F_{n})$. Duke Math. J. 168 (2019), no. 14, 2701--2726. doi:10.1215/00127094-2019-0022. https://projecteuclid.org/euclid.dmj/1568081121


Export citation

References

  • [1] Y. A. Bahturin, Identical Relations in Lie Algebras, VNU Science Press, Utrecht, 1987.
  • [2] M. R. Bridson and K. Vogtmann, Homomorphisms from automorphism groups of free groups, Bull. Lond. Math. Soc. 35 (2003), no. 6, 785–792.
  • [3] J. D. Dixon, M. Du Sautoy, A. Mann, and D. Segal, Analytic Pro-$p$ Groups, 2nd ed., Cambridge Stud. Adv. Math. 61, Cambridge Univ. Press, Cambridge, 1999.
  • [4] B. Farb and S. Hensel, Finite covers of graphs, their primitive homology, and representation theory, New York J. Math. 22 (2016), 1365–1391.
  • [5] B. Farb and S. Hensel, Moving homology classes in finite covers of graphs, Israel J. Math. 220 (2017), no. 2, 605–615.
  • [6] B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Math. Ser. 49, Princeton Univ. Press, Princeton, 2012.
  • [7] L. Funar, On the TQFT representations of the mapping class groups, Pacific J. Math. 188 (1999), no. 2, 251–274.
  • [8] W. Gaschütz, Über modulare Darstellungen endlicher Gruppend, die von freien Gruppen induziert werden, Math. Z. 60 (1954), 274–286.
  • [9] F. Grunewald, M. Larsen, A. Lubotzky, and J. Malestein, Arithmetic quotients of the mapping class group, Geom. Funct. Anal. 25 (2015), no. 5, 1493–1542.
  • [10] F. Grunewald and A. Lubotzky, Linear representations of the automorphism group of a free group, Geom. Funct. Anal. 18 (2009), no. 5, 1564–1608.
  • [11] A. Hadari, Every infinite order mapping class has an infinite order action on the homology of some finite cover, preprint, arXiv:1508.01555v1 [math.GT].
  • [12] A. Hadari, Homological eigenvalues of lifts of pseudo-Anosov mapping classes to finite covers, preprint, arXiv:1712.01416v1 [math.GT].
  • [13] A. Hadari, Non virtually solvable subgroups of mapping class groups have non virtually solvable representations, preprint, arXiv:1805.01527v1 [math.GT].
  • [14] M. Hall, Jr., Coset representations in free groups, Trans. Amer. Math. Soc. 67 (1949), 421–432.
  • [15] I. M. Isaacs, Finite Group Theory, Grad. Stud. Math. 92, Amer. Math. Soc., Providence, 2008.
  • [16] A. Kent, online communication, https://mathoverflow.net/q/86938 (accessed 13 August 2019).
  • [17] T. Koberda, Asymptotic linearity of the mapping class group and a homological version of the Nielsen-Thurston classification, Geom. Dedicata 156 (2012), no. 1, 13–30.
  • [18] T. Koberda, Alexander varieties and largeness of finitely presented groups, J. Homotopy Relat. Struct. 9 (2014), no. 2, 513–531.
  • [19] T. Koberda and R. Santharoubane, Quotients of surface groups and homology of finite covers via quantum representations, Invent. Math. 206 (2016), no. 2, 269–292.
  • [20] M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Éc. Norm. Supér. (4) 71 (1954), 101–190.
  • [21] Y. Liu, Virtual homological spectral radii for automorphisms of surfaces, preprint, arXiv:1710.05039v2 [math.GT].
  • [22] E. Looijenga, Prym representations of mapping class groups, Geom. Dedicata 64 (1997), no. 1, 69–83.
  • [23] E. Looijenga, Some algebraic geometry related to the mapping class group, conference lecture at “New Perspectives on the Interplay between Discrete Groups in Low-Dimensional Topology and Arithmetic Lattice,” Oberwolfach, 2015, available at https://www.staff.science.uu.nl/~looij101/.
  • [24] J. Marché, online communication, https://mathoverflow.net/q/86894 (accessed 13 August 2019).
  • [25] G. Masbaum, “An element of infinite order in TQFT-representations of mapping class groups” in Low-dimensional Topology (Funchal, 1998), Contemp. Math. 233, Amer. Math. Soc., Providence, 1999, 137–139.
  • [26] C. McMullen, Entropy on Riemann surfaces and the Jacobians of finite covers, Comment. Math. Helv. 88 (2013), no. 4, 953–964.
  • [27] A. Putman and B. Wieland, Abelian quotients of subgroups of the mappings class group and higher Prym representations, J. Lond. Math. Soc. (2) 88 (2013), no. 1, 79–96.
  • [28] J. J. Rotman, An Introduction to the Theory of Groups, 4th ed., Grad. Texts in Math. 148, Springer, New York, 1995.
  • [29] J.-P. Serre, Lie Algebras and Lie Groups, W. A. Benjamin, New York, 1965.
  • [30] J. R. Stallings, Topology of finite graphs, Invent. Math. 71 (1983), no. 3, 551–565.
  • [31] H. Sun, Virtual homological spectral radius and mapping torus of pseudo-Anosov maps, Proc. Amer. Math. Soc. 145 (2017), no. 10, 4551–4560.
  • [32] M. Suzuki, Geometric interpretation of the Magnus representation of the mapping class group, Kobe J. Math. 22 (2005), no. 1–2, 39–47.
  • [33] Svetlana, G. Robinson, and F. Ladisch, online communication, https://mathoverflow.net/q/232227.
  • [34] H. Zassenhaus, Ein Verfahren, jeder endlichenp-Gruppe einen Lie-Ring mit der Charakteristik $p$ zuzuordnen, Abh. Math. Sem. Univ. Hamburg 13 (1939), no. 1, 200–207.
  • [35] E. Zelmanov, “On some open problems related to the restricted Burnside problem” in Recent Progress in Algebra (Taejon/Seoul, 1997), Contemp. Math. 224, Amer. Math. Soc., Providence, 1999, 237–243.