[GAP Forum] Superperfect Group with Infinite Outer Automorphism Group?

[Jeffrey Rolland]

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Hello, all!

[I realize this is off-topic. I posted it to sci.math.research, but the
response rate there is sometimes poor, and I know you are all experts on
group theory. Please excuse the off-topic post.]

Recall that a group is perfect if and only if its abelianization is
trivial. Equivalently, P is perfect if and only if its first homology
group, H_1(P) = 0. A groups S is superperfect if and only if H_1(S) =
H_2(S) = 0. A finitely presented group is superperfect if it admits a
"balanced presentation", that is, a presentation with the same number of
generators and relators. The fundamental group of a homology sphere (a
manifold with the same integral homology of a sphere) is always
superperfect.

The automorphism group of a group is all of its automorphisms. The inner
automorphisms are a normal subgroup of the automorphism group, and the
outer automorphism group of a group is its automorphism group mod its
inner automorphism group.

Does anyone know of a finitely presented superperfect group (or, more
preferably, and (explicitly defined) infinite family of f.p.
superperfect groups) with infinite outer automorphism group? It would be
helpful if you could explicitly present the outer automorphism group (or
otherwise explicitly define it).

I only really have experience with one superperfect group, namely
SL(2,5), the special linear group in dimension two over the field with 5
elements. It's outer automorphism group is Z_2, about as far as you can
get from infinite.

Any assistance you can provide would be appreciated.

Sincerely,
- --
Jeffrey Rolland
<rollandj at uwm.edu>
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