The normal growth of linear groups over formal power serieses

TL;DR

The paper investigates the normal subgroup growth of the pro-$p$ group $\mathrm{SL}_2^1(R)$ and the growth of ideals in the power-series ring $R=\mathbb{F}_p[[x_1,\dots,x_d]]$ and in its Lie algebra $\mathfrak{sl}_2(R)$. It connects normal subgroups to ideals via congruence subgroups and uses Gröbner-basis methods to bound the number of generators of ideals, establishing that the growth types of $s_{p^k}^\triangleleft(R)$, $s_{p^k}^\triangleleft(\mathfrak{sl}_2(R))$, and $s_{p^k}^\triangleleft(\mathrm{SL}_2^1(R))$ are the same, with asymptotics $\log s_{p^k}^\triangleleft(R)\asymp k^{2-1/d}$. In the case $d=2$ they obtain a sharp leading term for the index-$p^N$ growth, $\log_p s_{p^N}^\triangleleft(R)=(2/3)^{3/2} N^{3/2}+O(N)$, confirming the optimality of the bounds in two variables. The results illuminate the contamination between polynomial and subexponential growth regimes and provide a robust counting framework via Gröbner bases for ideals in power-series rings, with implications for $p$-adic analytic groups and related subgroup-growth questions.

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 $R$.

Theorems & Definitions (24)

• Theorem 1: Segal
• Theorem 2
• Corollary 1
• Proposition 1
• Proposition 2
• proof
• Proposition 3
• proof
• Theorem 3: Buchberger
• Lemma 1