Genus stabilization for the homology of moduli spaces of orbit-framed curves with symmetries-I

TL;DR

The paper addresses genus stabilization for the homology of moduli spaces of curves with a fixed finite group action, including ramified cases. It develops a noncommutative ring of connected components $R$, builds Koszul-like $\mathcal{K}$-complexes from graded $R$-modules, and uses tethered-chains–based spectral sequences to control the $d^1$-differential, culminating in explicit stabilization ranges via an operator $U$. The main contributions are explicit Harer-type stabilization results: a precise threshold for $g'$ when $n=0$ and a generalization to arbitrary branch points $n$, with linear bounds on homology degrees in terms of invariants $\tilde{A}(R), A(R)$ and $A(M^n(0))$. The work significantly extends stabilization phenomena from unmarked moduli of curves to orbit-framed moduli with symmetries, linking Hurwitz monodromy data to homology stability and providing tools for computations in Hurwitz-type spaces.

Abstract

In a previous paper, arXiv:1301.4409, we showed that the moduli space of curves C with a G-symmetry (that is, with a faithful action of a finite group G), having a fixed generalized homological invariant, is irreducible if the genus g' of the quotient curve C' : = C/G satisfies g'>>0. Interpreting this result as stabilization for the 0-th homology group of the moduli space of curves with G-symmetry, we begin here a program for showing genus stabilization for all the homology groups of these spaces, in similarity to the results of Harer for the moduli space of curves. In this first paper we prove homology stabilization for a variant of the moduli space where one G-orbit is tangentially framed.

Figures (4)

• Figure 1: Attaching one handle endowed with a tethered chain. The base point for the fundamental group is any point on the red segment, equivalently, we take paths with endpoints in the segment.
• Figure 2: This collection of tethered chains yields a 3-simplex in the Hatcher-Vogtman complex, it determines a Riemann surface with boundary and a canonical isomorphism of its fundamental group with a free group of rank 8.
• Figure 3: If $a$ is the path on the left, and $b$ that on the right, both with the orientation given by the arrow, then they form a free basis of the fundamental group with respect to the base segment and the boundary oriented with the surface on the left is homotopic to the commutator $a b a^{-1} b^{-1}$
• Figure 4:

Theorems & Definitions (50)

• Theorem 1.1: Dunfield-Thurston
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 2.1
• Remark 2.2
• Definition 2.3
• Lemma 2.5
• proof
• Proposition 2.6