Timeline for Self-homomorphisms of surface groups
Current License: CC BY-SA 2.5
10 events
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| Apr 28, 2010 at 17:19 | comment | added | HJRW | The proof is easy enough to write in a comment. Suppose g is in the kernel of a surjection $\phi:G\to G$ and let $q: G\to Q$ be a finite quotient with $q(g)\neq 1$. Then it's easy to see that $q\circ\phi^n$ are all distinct maps $G\to Q$. But there are only finitely many maps from a finitely generated group to a fixed finite group. | |
| Apr 28, 2010 at 4:52 | history | edited | Andy Putman | CC BY-SA 2.5 |
added 3 characters in body
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| Apr 28, 2010 at 4:50 | comment | added | Lucas Culler | Thanks a lot! I'll take a look at the references next time I give up. I clearly have a lot to learn about group theory. | |
| Apr 28, 2010 at 4:32 | comment | added | Andy Putman | Good point Steve! | |
| Apr 28, 2010 at 4:31 | comment | added | Steve D | Not to nitpick, but all finitely generated residually finite groups are Hopfian. | |
| Apr 28, 2010 at 3:58 | comment | added | Andy Putman | And I should also point out that the proof in Lyndon and Schupp I referred to above is only one paragraph long and mostly self-contained. | |
| Apr 28, 2010 at 3:57 | vote | accept | Lucas Culler | ||
| Apr 28, 2010 at 3:57 | comment | added | Andy Putman | I gave a reference for a more general statement -- there's probably a more direct proof for surface groups, but the more general statement is extremely useful. For the residual finiteness of surface groups, see Hempel's beautiful 1-page paper "Residual finiteness of surface groups". | |
| Apr 28, 2010 at 3:54 | comment | added | Lucas Culler | I knew I could count on you. Should I just keep trying to prove it myself or is there a reference? | |
| Apr 28, 2010 at 3:49 | history | answered | Andy Putman | CC BY-SA 2.5 |