AbstractWe provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA).
The additional sufficient conditions are all met by finite groups, and so this case is fully characterised.
Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1].
for finite cyclic groups, as well as resolving the open case of that paper.
Let G be a group, and let r = r(t) be an element of the free product G * 〈G〉 of G with the infinite cyclic group generated by t. We say that the equation r(t) = 1 has a solution in G if the identity map on G extends to a homomorphism from G * 〈G〉 to G with r in its kernel. We say that r(t) = 1 has a solution over G if G can be embedded in a group H such that r(t) = 1 has a solution in H. This property is equivalent to the canonical map from G to 〈G, t|r〉 (the quotient of G * 〈G〉 by the normal closure of r) being injective.
The modular group PSL(2, ℤ), which is isomorphic to a free product of a cyclicgroupof order 2 and a cyclic group of order 3, has many important homomorphic images. Inparticular, Macbeath [7] showed that PSL(2, q) is an image of the modular group if q ≠ 9. (Here, as usual, q is a prime power.) The extended modular group PGL(2, ℤ) contains PSL{2, ℤ) with index 2. It has a presentationthe subgroup PSL(2, ℤ) being generated by UV and VW.
Amalgamated free product rigidity for group von Neumann algebras