Geometry & Topology

Fundamental groups of moduli stacks of stable curves of compact type

Marco Boggi

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Let ˜g,n, for 2g2+n>0, be the moduli stack of n–pointed, genus g, stable complex curves of compact type. Various characterizations and properties are obtained of both the topological and algebraic fundamental groups of the stack ˜g,n. For instance we show that the topological fundamental groups are linear, extending to all n0 previous results of Morita and Hain for g2 and n=0,1.

Let Γg,n, for 2g2+n>0, be the Teichmüller group associated with a compact Riemann surface of genus g with n points removed Sg,n, ie the group of homotopy classes of diffeomorphisms of Sg,n which preserve the orientation of Sg,n and a given order of its punctures. Let Kg,n be the normal subgroup of Γg,n generated by Dehn twists along separating simple closed curves (briefly s.c.c.) on Sg,n. The above theory yields a characterization of Kg,n for all n0, improving Johnson’s classical results for closed and one-punctured surfaces in [Topology 24 (1985) 113-126].

The Torelli group Tg,n is the kernel of the natural representation Γg,n Sp2g(). The abelianization of the Torelli group Tg,n is determined for all g1 and n1, thus completing classical results of Johnson [Topology 24 (1985) 127-144] and Mess [Topology 31 (1992) 775-790] for closed and one-punctured surfaces.

We also prove that a connected finite étale cover ˜λ of ˜g,n, for g2, has a Deligne–Mumford compactification ¯λ with finite fundamental group. This implies that, for g3, any finite index subgroup of Γg containing Kg has vanishing first cohomology group, improving a result of Hain [Math. Sci. Res. Inst. Publ. 28 (1995) 97-143].

Article information

Geom. Topol., Volume 13, Number 1 (2009), 247-276.

Received: 4 December 2007
Revised: 22 May 2008
Accepted: 9 September 2008
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 14H10: Families, moduli (algebraic) 30F60: Teichmüller theory [See also 32G15] 14F35: Homotopy theory; fundamental groups [See also 14H30]

Teichmüller group Torelli group


Boggi, Marco. Fundamental groups of moduli stacks of stable curves of compact type. Geom. Topol. 13 (2009), no. 1, 247--276. doi:10.2140/gt.2009.13.247.

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  • E Arbarello, M Cornalba, Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, J. Algebraic Geom. 5 (1996) 705–749
  • H Bass, J Milnor, J-P Serre, Solution of the congruence subgroup problem for ${\rm SL}\sb{n}\,(n\geq 3)$ and ${\rm Sp}\sb{2n}\,(n\geq 2)$, Inst. Hautes Études Sci. Publ. Math. (1967) 59–137
  • J S Birman, On Siegel's modular group, Math. Ann. 191 (1971) 59–68
  • M Boggi, Profinite Teichmüller theory, Math. Nachr. 279 (2006) 953–987
  • M Boggi, Monodromy of stable curves of compact type: rigidity and extension, Int. Math. Res. Not. IMRN (2007) 16pp Art. ID rnm017
  • M Boggi, M Pikaart, Galois covers of moduli of curves, Compositio Math. 120 (2000) 171–191
  • P Deligne, Le lemme de Gabber, Astérisque (1985) 131–150 Seminar on arithmetic bundles: the Mordell conjecture (Paris, 1983/84)
  • R Hain, Torelli groups and geometry of moduli spaces of curves, from: “Current topics in complex algebraic geometry (Berkeley, CA, 1992/93)”, Math. Sci. Res. Inst. Publ. 28, Cambridge Univ. Press (1995) 97–143
  • R Hain, E Looijenga, Mapping class groups and moduli spaces of curves, from: “Algebraic geometry–-Santa Cruz 1995”, Proc. Sympos. Pure Math. 62, Amer. Math. Soc. (1997) 97–142
  • J L Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. $(2)$ 121 (1985) 215–249
  • D Johnson, An abelian quotient of the mapping class group ${\cal I}\sb{g}$, Math. Ann. 249 (1980) 225–242
  • D Johnson, The structure of the Torelli group. I. A finite set of generators for ${\cal I}$, Ann. of Math. $(2)$ 118 (1983) 423–442
  • D Johnson, The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves, Topology 24 (1985) 113–126
  • D Johnson, The structure of the Torelli group. III. The abelianization of $\scr T$, Topology 24 (1985) 127–144
  • N Kawazumi, S Morita, The primary approximation to the cohomology of the moduli space of curves and cocycles for the Mumford–Morita–Miller classes, University of Tokio pre-print 2001–13 (2001)
  • F F Knudsen, The projectivity of the moduli space of stable curves. II. The stacks $M\sb{g,n}$, Math. Scand. 52 (1983) 161–199
  • I Madsen, M S Weiss, The stable moduli space of Riemann surfaces: Mumford's conjecture
  • G Mess, The Torelli groups for genus $2$ and $3$ surfaces, Topology 31 (1992) 775–790
  • S Mochizuki, Extending families of curves over log regular schemes, J. Reine Angew. Math. 511 (1999) 43–71
  • G Mondello, A remark on the homotopical dimension of some moduli spaces of stable Riemann surfaces, J. Eur. Math. Soc. $($JEMS$)$ 10 (2008) 231–241
  • S Morita, The extension of Johnson's homomorphism from the Torelli group to the mapping class group, Invent. Math. 111 (1993) 197–224
  • S Morita, A linear representation of the mapping class group of orientable surfaces and characteristic classes of surface bundles, from: “Topology and Teichmüller spaces (Katinkulta, 1995)”, World Sci. Publ., River Edge, NJ (1996) 159–186
  • B Noohi, Foundations of topological stacks I
  • B Noohi, Fundamental groups of algebraic stacks, J. Inst. Math. Jussieu 3 (2004) 69–103
  • M Pikaart, Moduli spaces of curves: stable cohomology and Galois covers, PhD thesis, Utrecht University (1997)
  • M Pikaart, A J de Jong, Moduli of curves with non-abelian level structure, from: “The moduli space of curves (Texel Island, 1994)”, Progr. Math. 129, Birkhäuser, Boston (1995) 483–509
  • L Ribes, P Zalesskii, Profinite groups, Ergebnisse der Math. und ihrer Grenzgebiete. 3. Folge. [Results in Math. and Related Areas. 3rd Series.] A Series of Modern Surveys in Math. 40, Springer, Berlin (2000)