Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Geometric representations of the braid groups   We show that the morphisms from the braid group with n strands in the mapping
class group of a surface with a possible non empty boundary, assuming that its
genus is smaller or equal to n/2 are either cyclic morphisms (their images are
cyclic groups), or transvections of monodromy morphisms (up to multiplication
by an element in the centralizer of the image, the image of a standard
generator of the braid group is a Dehn twist, and the images of two consecutive
standard generators are two Dehn twists along two curves intersecting in one
point). As a corollary, we determine the endomorphisms, the injective
endomorphisms, the automorphisms and the outer automorphism group of the
following groups: the braid group with n strands where n is greater than or
equal to 6, and the mapping class group of any surface of genus greater or
equal than 2. For each statement involving the mapping class group, we study
both cases: when the boundary is fixed pointwise, and when each boundary
component is fixed setwise. We will also describe the set of morphisms between
two different braid groups whose number of strands differ by at most one, and
the set of all morphisms between mapping class groups of surfaces (possibly
with boundary) whose genus (greater than or equal to 2) differ by at most one.

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The categories of this paper are: funct-an
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The categories of this paper are: atom-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: physics
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: comp-gas
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The categories of this paper are: alg-geom