Search results for " 20E36"
showing 2 items of 2 documents
Geometric représentations of the braid groups
2010
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 grou…
Embedding mapping class groups of orientable surfaces with one boundary component
2012
We denote by $S_{g,b,p}$ an orientable surface of genus $g$ with $b$ boundary components and $p$ punctures. We construct homomorphisms from the mapping class groups of $S_{g,1,p}$ to the mapping class groups of $S_{g',1,(b-1)}$, where $b\geq 1$. These homomorphisms are constructed from branched or unbranched covers of $S_{g,1,0}$ with some properties. Our main result is that these homomorphisms are injective. For unbranched covers, this construction was introduced by McCarthy and Ivanov~\cite{IM}. They proved that the homomorphisms are injective. A particular cases of our embeddings is a theorem of Birman and Hilden that embeds the braid group on $p$ strands into the mapping class group of …