largest subgroup of Out(^F2)Out(\hat{F_2}) which preserves the Nielsen invariant

Let x,y be generators for the free group F2. It’s known that Aut(F2), and hence Out(F2) preserves the conjugacy class of the subgroup [x,y] generated by [x,y] (This conjugacy class is in some contexts called the nielsen invariant)

On the other hand, if we view x,y as topological generators of ^F2 (hence fixing an embedding F2^F2), then Out(^F2) does not have the same property of preserving the conjugacy class of [x,y] (this follows from the fact that ^F2 has the strong lifting property). By the strong lifting property I mean that for any finite group G and two surjections φ,ψ:^F2G, there is an αAut(^F2) such that ψ=φα. If Aut(^F2) were to preserve the Nielsen invariant, then the image of the conjugacy class of [x,y] in G would be the same for any surjection ^F2G, which is to say that all commutators of generating pairs of finite groups generate conjugate subgroups. This is easily verified to be false – the smallest example is the alternating group A5 – it has two generating pair with commutators of order 3 and 5 respectively.

My question is – What is the stabilizer of the conjugacy class of [x,y] in Out(^F2)?

The stabilizer should be a closed subgroup, lets call it S – could it be ¯GL2(Z)¯Out(F2)^Out(F2)Out(^F2)?

If not, can we describe the difference S/¯GL2(Z) somehow?


Source : Link , Question Author : Will Chen , Answer Author : Community

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