Timeline for Automorphism group of a free product
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Nov 19, 2015 at 8:28 | comment | added | Thomas | Let us continue this discussion in chat. | |
| Nov 18, 2015 at 15:30 | comment | added | HJRW | @Thomas, I'm really not sure what point you're trying to make. I stand by my assertion that, when $k=0$, the picture is 'relatively simple'. You assert that the case of three free factors is 'complicated'. The upshot of this discussion doesn't seem very interesting except to conclude that what I think is 'relatively simple' you think is 'complicated'. | |
| Nov 18, 2015 at 15:17 | comment | added | Thomas | @HRW, but it does not if you have 3 factors $A*B*C*$ and you conjugate $A$, not $C by some $b$... They are analogs of Dehn twists. These animals generate the so called "Fuchs-Rabinowitz group", which has completely be describe in the 40's, I think. They are usefull to describe the mapping class group of connected sum in dim 3, where they play the role of Dehn twists. | |
| Nov 18, 2015 at 15:13 | comment | added | HJRW | ... Anyway, as I tried to make clear, the point of my answer was not to give a complete description of the automorphism group (there are perfectly good references in the literature), but to point out that in order to give such a description, one should consider the Grushko decomposition, and not just the free splitting that one has in hand. | |
| Nov 18, 2015 at 15:09 | comment | added | HJRW | @Thomas, on the contrary, it does 'come from ... $\mathrm{Aut}(B)$'. More precisely, it's the composition of an inner automorphism of $B$ (which, as you rightly observe, gives you a non-trivial outer automorphism of the free product) with an inner automorphism of $A*B$. I would classify such things as not 'complicated', though I concede that what one may find complicated is a matter of taste. (But the definition I have in mind is that an automorphism is 'not complicated' if it fixes a point in the deformation space of Grushko decompositions.) ... | |
| Nov 18, 2015 at 15:01 | comment | added | Thomas | It is an automorphism which is not inner, and does not come from $Aut(A)$, $Aut(B)$, an analog of Dehn twist. In fact the automorphism group of $A*B$ is generated by such automorphisms, $Aut(A), Aut(B)$, and inner automorphisms....In the case of $A*Z$, you must add the automorphism which is $Id$ on $A$ and map $t$ to $ta$ if $t$ is the generator of $Z$. The general case is obtained by mixing these sorts of automorphisms... | |
| Nov 18, 2015 at 13:01 | comment | added | HJRW | @Thomas, well, it depends what you mean by 'complicated'. | |
| Nov 18, 2015 at 12:06 | comment | added | Thomas | In fact, even with only two non free factors, $A,B$ you have complicated automorphism called "partial conjugations" : let $b\in B$, then there exists an automorphism which fixes $B$ and conjugate $A$ to $bAb^{-1}$ | |
| Nov 18, 2015 at 10:24 | comment | added | Ofra | Excellent, thank you! Actually your answer is more then what i was hoping for ! | |
| Nov 18, 2015 at 10:22 | vote | accept | Ofra | ||
| Nov 18, 2015 at 10:20 | history | answered | HJRW | CC BY-SA 3.0 |