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Arithmetic representations of mapping class groups

Eduard Looijenga

TL;DR

This work analyzes the G-equivariant symplectic representation of the mapping class group on the first homology of a surface, focusing on the image inside $ ext{Sp}( ext{H}_1(S))^G$ for a finite group $G$ acting on a genus $ ext{h}$ surface. The author decomposes $ ext{H}_1(S)$ into isotypical ${b Q}G$-summands and studies each factor via associated unitary groups $ ext{U}(H[ ext{χ}])$, proving arithmeticity under broad conditions and identifying low-rank exceptions. A central tool is an arithmeticity criterion for transvection-generated subgroups in unitary groups, established by induction on the dimension of the hyperbolic modules and using Eichler transformations to propagate lattices. Topologically, liftable $G$-equivariant mapping classes are constructed explicitly via push maps and Dehn twists, ensuring large transvection-generated subgroups appear in the symplectic image; this yields the Putman–Wieland property after stabilization by a genus-one or genus-two base. The results extend and complement prior work on arithmetic quotients of mapping class groups, providing concrete, constructive routes to arithmetic images in each isotypical factor.

Abstract

Let $S$ be a closed oriented surface and $G$ a finite group of orientation preserving automorphisms of $S$ whose orbit space has genus at least $2$. There is a natural group homomorphism from the $G$-centralizer in $Diff^+(S)$ to the $G$-centralizer in $Sp(H_1(S))$. We give a sufficient condition for its image to be a subgroup of finite index and a weaker condition for this to have no finite nonzero orbit (the Putman-Wieland property).

Arithmetic representations of mapping class groups

TL;DR

This work analyzes the G-equivariant symplectic representation of the mapping class group on the first homology of a surface, focusing on the image inside for a finite group acting on a genus surface. The author decomposes into isotypical -summands and studies each factor via associated unitary groups , proving arithmeticity under broad conditions and identifying low-rank exceptions. A central tool is an arithmeticity criterion for transvection-generated subgroups in unitary groups, established by induction on the dimension of the hyperbolic modules and using Eichler transformations to propagate lattices. Topologically, liftable -equivariant mapping classes are constructed explicitly via push maps and Dehn twists, ensuring large transvection-generated subgroups appear in the symplectic image; this yields the Putman–Wieland property after stabilization by a genus-one or genus-two base. The results extend and complement prior work on arithmetic quotients of mapping class groups, providing concrete, constructive routes to arithmetic images in each isotypical factor.

Abstract

Let be a closed oriented surface and a finite group of orientation preserving automorphisms of whose orbit space has genus at least . There is a natural group homomorphism from the -centralizer in to the -centralizer in . We give a sufficient condition for its image to be a subgroup of finite index and a weaker condition for this to have no finite nonzero orbit (the Putman-Wieland property).

Paper Structure

This paper contains 5 sections, 15 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction and statement of the main result
  2. Brief review of special unitary groups
  3. The hyperbolic module attached to a finite group
  4. An arithmeticity criterion
  5. Finding liftable mapping classes

Key Result

Theorem 1.1

Let $S\to S_G$ be a $G$-cover as above.

Figures (2)

  • Figure 1: The surface $S_G$, its quotient $S_G(\alpha)$ and the normalization $\widehat{S}_G(\alpha)$.
  • Figure 2: Point pushing (or rather, 'small circle pushing') the genus one surface $S^\beta$ along $S_G/S^\beta_G$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 3.1
  • Remark 3.2
  • Theorem 3.3: Raghunathan raghunathan, Venkataramana venky
  • ...and 23 more