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Higher-order signature cocycles for subgroups of the mapping class
group

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>Peter Horn (Columbia University)

Abstract: This is joint work with Tim Cochran and Shelly Harvey. We define
families of invariants for elements of the mapping class group of $\Sigma$,
a compact, orientable surface. For a characteristic subgroup
$H\triangleleft\pi_1\Sigma$, let $J(H)$ denote the subgroup of mapping
classes that induce the identity map on $\pi_1\Sigma/H$. To a unitary
representation $\psi$ of $\pi_1\Sigma/H$, we associate a higher-order
$\rho$-invariant, $\rho_\psi$, and a signature 2-cocycle $\sigma_\psi$, a
generalization of the Meyer cocycle. We show that each $\rho_\psi$ is a
quasimorphism from $J(H)\to\mathbb{R}$, and that the $\sigma_\psi$ span an
infinite rank subgroup of $H^2_{bounded}(J(H);\mathbb{R})$