Abstract. We prove that for each sufficiently complicated orientable surface S , there exists an infinite image linear representation ρ of π 1 (S ) such that if γ ∈ π 1 (S ) is freely homotopic to a simple closed curve on S , then ρ(γ) has finite order. Furthermore, we prove that given a sufficiently complicated orientable surface S , there exists a regular finite cover S → S such that H 1 (S , Z) is not generated by lifts of simple closed curves on S , and we give a lower bound estimate on the index of the subgroup generated by lifts of simple closed curves. We thus answer two questions posed by Looijenga, and independently by Kent, Kisin, Marché, and McMullen. The construction of these representations and covers relies on quantum SO(3) representations of mapping class groups.
For N ≥ 2, we study a certain sequence (ρ (cp) p ) of N-dimensional representations of the mapping class group of the one-holed torus arising from SO(3)-TQFT, and show that the conjecture of Andersen, Masbaum, and Ueno [1] holds for these representations. This is done by proving that, in a certain basis and up to a rescaling, the matrices of these representations converge as p tends to infinity. Moreover, the limits describe the action of SL2(Z) on the space of homogeneous polynomials of two variables of total degree N − 1.
We prove that the Witten-Reshetikhin-Turaev SU(2) quantum representations of mapping class groups are always irreducible in the case of surfaces equipped with colored banded points, provided that at least one banded point is colored by one. We thus generalize a well-known result due to J. Roberts.for the quantum representation of Mod(S , n) arising from Witten-Reshetikhin-Turaev
Let Σ be a surface with negative Euler characteristic, genus at least one and at most one boundary component. We prove that the Kauffman bracket skein algebra of Σ over the field of rational functions can be algebraically generated by a finite number of simple closed curves that are naturally associated to certain generators of the mapping class group of Σ. The action of the mapping class group on the skein algebra gives canonical relations between these generators. From this, we conjecture a presentation for a skein algebra of Σ.
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