We propose a family of homological representations of the braid groups on surfaces. This family extends linear representations of the braid groups on a disc, such as the Burau representation and the Lawrence-KrammerBigelow representation.
In this paper, we compute the automorphism groups [Formula: see text] and [Formula: see text] of braid groups [Formula: see text] and [Formula: see text] on every orientable surface [Formula: see text], which are isomorphic to group extensions of the extended mapping class group [Formula: see text] by the transvection subgroup except for a few cases. We also prove that [Formula: see text] is always a characteristic subgroup of [Formula: see text], unless [Formula: see text] is a twice-punctured sphere and [Formula: see text].
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