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Information Center for Mathematical Science

논문검색

Information Center for Mathematical Science

논문검색

Duke Mathematical Journal
( Vol. 117 NO.2 / (2003))
Congruence Subgroup Growth of Arithmetic Groups in Positive Characteristic
Miklos Abert, Nikolay Nikolov, Balazs Szegedy,
Pages. 367-383
Abstract 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 $SL_n (F_p[t]) (n geq 3)$ is of type $n^{log , n}$. 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 sub-algebras of $L$.
Contents 1. Introduction 2. The global case: $X(O_S)$ 3. The local case: $G(1)$ and its graded Lie algebra 4. Rings of higher Krull dimension 5. Subspaces of Chevalley Lie algebras
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Mathmatical Subject Classification 20E07, 20H05, 17B70