On sequences of word maps of compact topological groups

N. L. Gordeev

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 35, Tome 492 (2020), pp. 94-124

Résumé

In the paper of A. Thom (A. Thom, Convergent sequences in discrete groups, Canad. Math. Bull. 56 (2013), 424–433) it has been proved that for any standard unitary group $\mathrm{SU}(\mathbb{C})$ (the compact form) and for any real number $\epsilon > 0$ there is a non-trivial word $w(x, y)$ on two variables such that the image of the word map $\tilde{w}: \mathrm{SU}_n(\mathbb{C})^2\rightarrow \mathrm{SU}_n (\mathbb{C})$ is contained in $\epsilon$-neighbourhood of the identity of the group $\mathrm{SU}_n(\mathbb{C})$. Actually, in Thom's paper there is a construction of a sequence $\{w_j\}_{j \in \mathbb{N}}$, where $w_j \in F_2$, that converges uniformly on a compact group to the identity. In this paper we propose a method for the construction of such sequences. Also, using the result of T. Bandman, G-M. Greuel, F. Grunewald, B. Kunyavskii, G. Pfister and E. Plotkin, Identities for finite solvable groups and equations in finite simple groups. – Compositio Math. 142 (2006) 734-764), we construct the sequence of the surjective word maps $\tilde{w}_j: \mathrm{SU}_2(\mathbb{C})^n\rightarrow \mathrm{SU}_2(\mathbb{C})$, where each word $w_j$ is contained in the corresponding member $F_n^j$ of the derived series of the free group $F_n$. We also make some comments and remarks which are relevant to such results and to general properties of word maps of compact groups.