On the group of substitutions of formal power series with integer coefficients

I. K. Babenko ; S. A. Bogatyi

Izvestiya. Mathematics , Tome 72 (2008) no. 2, pp. 241-264

Résumé

We study certain properties of the group $\mathcal J(\mathbb Z)$ of substitutions of formal power series in one variable with integer coefficients. We show that $\mathcal J(\mathbb Z)$, regarded as a topological group, has four generators and cannot be generated by fewer elements. In particular, we show that the one-dimensional continuous homology of $\mathcal J(\mathbb Z)$ is isomorphic to $\mathbb Z\oplus\mathbb Z\oplus\mathbb Z_2\oplus\mathbb Z_2$. We study various topological and geometric properties of the coset space $\mathcal J(\mathbb R)/\mathcal J(\mathbb Z)$. We compute the real cohomology $\widetilde{H}^*\bigl(\mathcal J(\mathbb Z); \mathbb R\bigr)$ with uniformly locally constant supports and show that it is naturally isomorphic to the cohomology of the nilpotent part of the Lie algebra of formal vector fields on the line.