The normal growth of linear groups over formal power serieses

TL;DR

This paper investigates the structure and enumeration of normal subgroups and ideals within certain linear groups and associated algebraic structures over formal power series rings, providing estimates and counts for these algebraic objects.

Contribution

It offers new estimates and counts for normal subgroups of SL_2 over formal power series rings and for ideals in related Lie and associative algebras.

Findings

Estimated the number of normal subgroups of SL_2 over formal power series rings.

Counted the number of ideals in the Lie algebra of R.

Counted the number of ideals in the associative algebra R.

Abstract

Put $R=\F[[t_1, \ldots, t_d]])$. We estimate the number of normal subgroups of $\mathrm{SL}_2^1(\F[[t_1, \ldots, t_d]])$ for $p>2$, the number of ideals in the Lie algebra $\Lie(R)$, and the number of ideals in the associative algebra $R$.