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Hyperelliptic graphs and the period mapping on outer space. (English) Zbl 1494.20058

Summary: The period mapping assigns to each rank \(n\), marked metric graph {\(\Gamma\)} a positive definite quadratic form on \(H_1(\Gamma,\mathbb{R})\). This defines maps \(\overline{\Phi}\) and \(\Phi\) on Culler-Vogtmann’s outer space \(CV_n\), and its Torelli space quotient \(\mathcal{T}_n\), respectively. The map {\(\Phi\)} is a free group analog of the classical period mapping that sends a marked Riemann surface to its Jacobian. In this paper, we analyze the fibers of \(\Phi\) in \(\mathcal{T}_n\), showing that they are aspherical, \(\pi_1\)-injective subspaces. Metric graphs admitting a hyperelliptic involution play an important role in the structure of {\(\Phi\)}, leading us to define the hyperelliptic Torelli group, \(\mathcal{ST}(n)\le \mathrm{Out}(F_n)\). We obtain generators for \(\mathcal{ST}(n)\), and apply them to show that the connected components of the locus of ‘hyperelliptic’ graphs in \(\mathcal{T}_n\) become simply connected when certain degenerate graphs at infinity are added.

MSC:

20F65 Geometric group theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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[1] 1 N. A’Campo, ‘Tresses, monodromie et le groupe symplectique’, Comment. Math. Helv.54 (1979) 318-327. · Zbl 0441.32004
[2] 2 J. Aramayona and J. Souto, ‘Automorphisms of the graph of free splittings’, Michigan Math. J.60 (2011) 483-493. · Zbl 1242.05117
[3] 3 V. I. Arnol’d, ‘A remark on the branching of hyperelliptic integrals as functions of the parameters’, Funct. Anal. Appl.2 (1968) 1-3.
[4] 4 O. Baker, ‘The Jacobian map on Outer space’, PhD Thesis, Cornell University, Ithaca, NY, 2011.
[5] 5 H. Bass, M. Lazard and J.‐P. Serre, ‘Sous‐groupes d’indice fini dans SL(n,Z)’, Bull. Amer. Math. Soc.70 (1964) 385-392. · Zbl 0232.20086
[6] 6 M. Bestvina, K.‐U. Bux and D. Margalit, ‘Dimension of the Torelli group for Out (F n)’, Invent. Math.170 (2007) 1-32. · Zbl 1135.20026
[7] 7 T. Brendle, D. Margalit and A. Putman, ‘Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=−1’, Invent. Math.200 (2015) 263-310. · Zbl 1328.57021
[8] 8 M. R. Bridson and K. Vogtmann, ‘On the geometry of the automorphism group of a free group’, Bull. Lond. Math. Soc.27 (1995) 544-552. · Zbl 0836.20045
[9] 9 L. Caporaso and F. Viviani, ‘Torelli theorem for stable curves’, J. Eur. Math. Soc.13 (2011) 1289-1329. · Zbl 1230.14037
[10] 10 M. Chan, ‘Combinatorics of the tropical Torelli map’, Algebra Number Theory6 (2012) 1133-1169. · Zbl 1283.14028
[11] 11 M. Chan, ‘Tropical hyperelliptic curves’, J. Algebraic Combin.37 (2013) 331-359. · Zbl 1266.14050
[12] 12 M. Chan, S. Galatius and S. Payne, ‘The tropicalization of the moduli space of curves II: Topology and applications’, Preprint, 2016, http://arxiv.org/pdf/1604.03176.
[13] 13 D. J. Collins, ‘Cohomological dimension and symmetric automorphisms of a free group’, Comment. Math. Helv.64 (1989) 44-61. · Zbl 0669.20027
[14] 14 D. J. Collins, ‘Palindromic automorphisms of free groups’, Combinatorial and geometric group theory (Edinburgh, 1993), London Mathematical Society Lecture Note Series 204 (Cambridge University Press, Cambridge, 1995) 63-72. · Zbl 0843.20022
[15] 15 M. Culler, ‘Finite groups of outer automorphisms of a free group’, Contributions to group theory, Contemporary Mathematics 33 (American Mathematical Society, Providence, RI, 1984) 197-207. · Zbl 0552.20024
[16] 16 M. Culler and K. Vogtmann, ‘Moduli of graphs and automorphisms of free groups’, Invent. Math.84 (1986) 91-119. · Zbl 0589.20022
[17] 17 D. I. Fouxe‐Rabinovitch, ‘Über die Automorphismengruppen der freien Produkte. I’, Rec. Math. [Mat. Sbornik] N.S.8 (1940) 265-276. · JFM 66.0066.04
[18] 18 D. I. Fouxe‐Rabinovitch, ‘Über die Automorphismengruppen der freien Produkte. II’, Rec. Math. [Mat. Sbornik] N.S.9 (1941) 183-220. · Zbl 0025.00904
[19] 19 N. J. Fullarton, ‘A generating set for the palindromic Torelli group’, Algebr. Geom. Topol.15 (2015) 3535-3567. · Zbl 1368.20026
[20] 20 N. D. Gilbert, ‘Presentations of the automorphism group of a free product’, Proc. Lond. Math. Soc. (3) 54 (1987) 115-140. · Zbl 0609.20023
[21] 21 H. H. Glover and C. A. Jensen, ‘Geometry for palindromic automorphism groups of free groups’, Comment. Math. Helv.75 (2000) 644-667. · Zbl 0972.20021
[22] 22 J. T. Griffin, ‘Diagonal complexes and the integral homology of the automorphism group of a free product’, Proc. Lond. Math. Soc. (3) 106 (2013) 1087-1120. · Zbl 1278.20070
[23] 23 S. Grushevsky, ‘The Schottky problem’, Current developments in algebraic geometry, Mathematical Sciences Research Institute Publications 59 (Cambridge University Press, Cambridge, 2012) 129-164. · Zbl 1254.14054
[24] 24 M. Handel and L. Mosher, ‘The free splitting complex of a free group’, I: hyperbolicity’, Geom. Topol.17 (2013) 1581-1672. · Zbl 1278.20053
[25] 25 A. Hatcher, ‘Homological stability for automorphism groups of free groups’, Comment. Math. Helv.70 (1995) 39-62. · Zbl 0836.57003
[26] 26 C. Jensen, J. McCammond and J. Meier, ‘The Euler characteristic of the Whitehead automorphism group of a free product’, Trans. Amer. Math. Soc.359 (2007) 2577-2595. · Zbl 1124.20037
[27] 27 R. Kobayashi, ‘A finite presentation of the level 2 principal congruence subgroup of G L(n;Z)’, Kodai Math. J.38 (2015) 534-559. · Zbl 1332.57004
[28] 28 M. Kotani and T. Sunada, ‘Jacobian tori associated with a finite graph and its abelian covering graphs’, Adv. Appl. Math.24 (2000) 89-110. · Zbl 1017.05038
[29] 29 S. Krstić, ‘Finitely generated virtually free groups have finitely presented automorphism group’, Proc. Lond. Math. Soc. (3) 64 (1992) 49-69. · Zbl 0773.20008
[30] 30 J. L. Mennicke, ‘Finite factor groups of the unimodular group’, Ann. of Math. (2) 81 (1965) 31-37. · Zbl 0135.06504
[31] 31 G. Mikhalkin, ‘Tropical geometry and its applications’, International Congress of Mathematicians, vol. II (European Mathematical Society, Zürich, 2006) 827-852. · Zbl 1103.14034
[32] 32 G. Mikhalkin and I. Zharkov, ‘Tropical curves, their Jacobians and theta functions’, Curves and abelian varieties, Contemporary Mathematics 465 (American Mathematical Society, Providence, RI, 2008) 203-230. · Zbl 1152.14028
[33] 33 T. Nagnibeda, ‘The Jacobian of a finite graph’, Harmonic functions on trees and buildings (New York, 1995), Contemporary Mathematics 206 (American Mathematical Society, Providence, RI, 1997) 149-151. · Zbl 0883.05069
[34] 34 D. Quillen, ‘Higher algebraic K‐theory. I’, Algebraic K‐theory, I: higher K‐theories, Proceedings of the Conference Held at the Seattle Research Center of Battelle Memorial Institute, Seattle, Washington, 1972, Lecture Notes in Mathematics 341 (Springer, Berlin, 1973) 85-147. · Zbl 0292.18004
[35] 35 F. Vallentin, ‘Sphere covering, lattices, and tilings (in low dimensions)’, Dissertation, Technische Universität München, München, 2003.
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