Inner? Automorphisms - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:43:41Z http://mathoverflow.net/feeds/question/99250 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99250/inner-automorphisms Inner? Automorphisms Igor Rivin 2012-06-10T18:19:17Z 2012-06-11T04:19:16Z <p>It is a fairly easy result of Edna Grossman's that any automorphism of a (finitely generated) free group which acts by conjugation on every primitive element ( primitive element in$F_n$ is one which is a member of a generating set of order $n$) is in fact inner. The question is: what is the right generalization of this fact? For surface groups, one can replace primitive elements by simple curves, but I a, not sure if that the"canonical" thing to do...</p> <p><strong>UPDATE</strong> Looking up citations of Henry's (@HW's) great reference, I found that the paper he cites is not quite "the last work". A more recent (and very relevant) word seems to be Bogopolski-Ventura (On endomorphisms of torsion-free hyperbolic groups) from which I quote the abstract:</p> <p>Let $H$ be a torsion-free $\delta$-hyperbolic group with respect to a finite generating set $S$. Let $a_1,..., a_n$ and $a_{1*},..., a_{n*}$ be elements of $H$ such that $a_{i*}$ is conjugate to $a_i$ for each $i=1,..., n$. Then, there is a uniform conjugator if and only if $W(a_{1*},..., a_{n*})$ is conjugate to $W(a_1,..., a_n)$ for every word $W$ in $n$ variables and length up to a computable constant depending only on $\delta$, $\sharp{S}$ and $\sum_{i=1}^n |a_i|$. As a corollary, we deduce that there exists a computable constant $\mathcal{C}=\mathcal{C}(\delta, \sharp S)$ such that, for any endomorphism $\phi$ of $H$, if $\phi(h)$ is conjugate to $h$ for every element $h\in H$ of length up to $\mathcal {C}$, then $\phi$ is an inner automorphism. Another corollary is the following: if $H$ is a torsion-free conjugacy separable hyperbolic group, then $\mbox{Out}(H)$ is residually finite. When particularizing the main result to the case of free groups, we obtain a solution for a mixed version of the classical Whitehead's algorithm.</p> http://mathoverflow.net/questions/99250/inner-automorphisms/99262#99262 Answer by HW for Inner? Automorphisms HW 2012-06-10T22:11:16Z 2012-06-10T22:11:16Z <p>Have you looked at the following paper of Minasyan and Osin, who generalize <em>Edna</em> Grossman's Theorem to the relatively hyperbolic case?</p> <p>Minasyan, A.(4-SHMP-SM); Osin, D.(1-VDB) Normal automorphisms of relatively hyperbolic groups. <em>Trans. Amer. Math. Soc.</em> 362 (2010), no. 11, 6079–6103.</p> <p>It's the state of the art for these questions, I believe.</p>