Fundamental groups of moduli stacks of stable curves of compact type

Abstract

LetM g;n , for 2g 2 C 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 M g;n . For instance, in Theorem 3.20, we show that the topological fundamental groups are linear, extending to all n 0 previous results of Morita and Hain for g 2 and n D 0; 1. Let g;n , for 2g 2 C n > 0, be the Teichmüller group associated with a compact Riemann surface of genus g with n points removed S g;n , ie the group of homotopy classes of diffeomorphisms of S g;n which preserve the orientation of S g;n and a given order of its punctures. Let K g;n be the normal subgroup of g;n generated by Dehn twists along separating simple closed curves (briefly s.c.c.) on S g;n . The above theory yields a characterization of K g;n for all n 0, improving Johnson's classical results for closed and one-punctured surfaces in [13].The Torelli group T g;n is the kernel of the natural representation g;n ! Sp 2g .Z/. The abelianization of the Torelli group T g;n is determined for all g 1 and n 1, thus completing classical results of Johnson [14] and Mess [18] for closed and one-punctured surfaces.We also prove that a connected finite étale cover M of M g;n , for g 2, has a Deligne-Mumford compactification S M with finite fundamental group. This implies that, for g 3, any finite index subgroup of g containing K g has vanishing first cohomology group, improving a result of Hain [8].

A note on the abelianizations of finite-index subgroups of the mapping class group

“…As a corollary, we obtain the following result, which was recently proven by Boggi [2] via a difficult algebro-geometric argument under the assumption b = p = 0.…”

Abelian quotients of subgroups of the mapping class group and higher Prym representations

“…In [12], Hain verifies this for Γ that contain the Torelli group, which is the kernel of the action of Mod p g,n on H 1 (Σ 0 g,0 ; Z). This was later generalized to some deeper subgroups by Boggi [6] and the first author [22].…”

“…. , η k of simple closed nonseparating curves on Σ p g,n such that γ = η 1 , such that γ ′ = η k , and such that η i and η i+1 intersect exactly once for 1 ≤ i < k. To show that the images of the maps in (6) are the same, it is thus enough to show that the images of the maps…”

Genus 3 mapping class groups are not Kähler

“…are the weight 4 2 and 3 components of the Chevalley-Eilenberg chains of n. Here J is the "Jacobi identity" map…”