%0 Generic %1 Aramayona2010 %A Aramayona, Javier %A Souto, Juan %D 2010 %K groups ptolemy surfaces teichmuller %T Homomorphisms between mapping class groups %U http://arxiv.org/abs/1011.1855 %X Suppose that $X$ and $Y$ are surfaces of finite topological type, where $X$ has genus $g6$ and $Y$ has genus at most $2g-1$; in addition, suppose that $Y$ is not closed if it has genus $2g-1$. Our main result asserts that every non-trivial homomorphism $\Map(X) \Map(Y)$ is induced by an \em embedding, i.e. a combination of forgetting punctures, deleting boundary components and subsurface embeddings. In particular, if $X$ has no boundary then every non-trivial endomorphism $\Map(X)\to\Map(X)$ is in fact an isomorphism. As an application of our main theorem we obtain that, under the same hypotheses on genus, if $X$ and $Y$ have finite analytic type then every non-constant holomorphic map $\CM(X)\to\CM(Y)$ between the corresponding moduli spaces is a forgetful map. In particular, there are no such holomorphic maps unless $X$ and $Y$ have the same genus and $Y$ has at most as many marked points as $X$.

@misc{Aramayona2010, abstract = { Suppose that $X$ and $Y$ are surfaces of finite topological type, where $X$ has genus $g\geq 6$ and $Y$ has genus at most $2g-1$; in addition, suppose that $Y$ is not closed if it has genus $2g-1$. Our main result asserts that every non-trivial homomorphism $\Map(X) \to \Map(Y)$ is induced by an {\em embedding}, i.e. a combination of forgetting punctures, deleting boundary components and subsurface embeddings. In particular, if $X$ has no boundary then every non-trivial endomorphism $\Map(X)\to\Map(X)$ is in fact an isomorphism. As an application of our main theorem we obtain that, under the same hypotheses on genus, if $X$ and $Y$ have finite analytic type then every non-constant holomorphic map $\CM(X)\to\CM(Y)$ between the corresponding moduli spaces is a forgetful map. In particular, there are no such holomorphic maps unless $X$ and $Y$ have the same genus and $Y$ has at most as many marked points as $X$. }, added-at = {2010-11-11T07:45:59.000+0100}, author = {Aramayona, Javier and Souto, Juan}, biburl = {https://www.bibsonomy.org/bibtex/2c346fd9bb5636509f20113eea88d8581/uludag}, description = {Homomorphisms between mapping class groups}, interhash = {46403d063099e6eb97fc0559acf76ebc}, intrahash = {c346fd9bb5636509f20113eea88d8581}, keywords = {groups ptolemy surfaces teichmuller}, note = {cite arxiv:1011.1855 }, timestamp = {2010-11-11T07:45:59.000+0100}, title = {Homomorphisms between mapping class groups}, url = {http://arxiv.org/abs/1011.1855}, year = 2010 }