Let $H$ be a finite index subgroup of a finitely generated group $G$. Assume that $Out(H)$ is finite. Can $Out(G)$ be infinite?


I think the answer is yes. Let $A$ be a f.g. group. Assume that $\text{Out}(A)=1$ ($\text{Out}(A)$ finite would probably be enough) and that $A$ contains a central copy of $C^{(\infty)}$, the direct sum of countably many copies of the cyclic group $C$ of prime order $p$. Then $\text{Out}(A\times C)$ is infinite: indeed if $f$ is a homomorphism from $C$ to the center of $A$, then if we set $u_f(a,t)=(af(t),t)$, then $f\mapsto u_f$ injects $\text{Hom}(C,Z(A))$ into $\text{Out}(A\times C)$. Actually all these automorphisms are identity on the finite index subgroup $A$, and this is essentially the reason I don't expect Mark's argument to work.

It remains to find an example of such a group $A$. I expect some complicated nilpotent-by-abelian group over $\mathbf{F}_p[t]$ to work (or Kac-Moody groups??), but I'm not prone to go into computations and maybe somebody else has one example.

Edit: Anton Klyachko's answer to my subsequent question confirms the existence of such a group $A$. Therefore, the answer to the question here is positive:

for finitely generated groups, having infinite outer automorphism group does not pass to finite index subgroups.

  • $\begingroup$ There is a nice result by Ould Houcine: every countable abelian group injects into the centre of some finitely presented group: math.univ-lyon1.fr/~ould/Fiches-papiers/embeddings.pdf $\endgroup$ – Alain Valette May 1 '11 at 20:47
  • $\begingroup$ Yes but there are many classical ways to inject an infinite-dimensional $\mathbf{F}_p$-vector space into the center of a f.p. group (e.g. into Abels' group over $\mathbf{F}_p$). But we need it to have finite Out. I think it should work by picking some suitable f.g. $\mathbf{F}_p$-algebra $A$ with a finite automorphism group, with an infinite $K_2$. Probably some localization of $\mathbf{F}_p[u,v]$ should work. Then the universal central extension of $\mathbf{SL}_d(A)$ should be the example. --YC $\endgroup$ – YCor May 2 '11 at 11:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.