Duke Mathematical Journal

Congruence subgroup growth of arithmetic groups in positive characteristic

Miklós Abért, Nikolay Nikolov, and Balázs Szegedy

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We prove a new uniform bound for subgroup growth of a Chevalley group $G$ over the local ring $\mathbb {F}[[t]]$ and also over local pro-$p$ rings of higher Krull dimension. This is applied to the determination of congruence subgroup growth of arithmetic groups over global fields of positive characteristic. In particular, we show that the subgroup growth of ${\rm SL}\sb n(F\sb p[t]) (n\geq3)$ is of type $n\sp {\log n}$. This was one of the main problems left open by A. Lubotzky in his article [5].

The essential tool for proving the results is the use of graded Lie algebras. We sharpen Lubotzky's bounds on subgroup growth via a result on subspaces of a Chevalley Lie algebra $L$ over a finite field $\mathbb {F}$. This theorem is proved by algebraic geometry and can be modified to obtain a lower bound on the codimension of proper Lie subalgebras of $L$.

Article information

Duke Math. J., Volume 117, Number 2 (2003), 367-383.

First available in Project Euclid: 26 May 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20H05: Unimodular groups, congruence subgroups [See also 11F06, 19B37, 22E40, 51F20]
Secondary: 17B45: Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx] 20G30: Linear algebraic groups over global fields and their integers


Abért, Miklós; Nikolov, Nikolay; Szegedy, Balázs. Congruence subgroup growth of arithmetic groups in positive characteristic. Duke Math. J. 117 (2003), no. 2, 367--383. doi:10.1215/S0012-7094-03-11726-3. https://projecteuclid.org/euclid.dmj/1085598374

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