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Subgroup growth in some pro- groups

Authors:	Yiftach Barnea Robert Guralnick

Institution:	Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706 ; Department of Mathematics, University of Southern California, Los Angeles, California 90089-1113

Abstract:	For a group let $a_{n}(G)$ be the number of subgroups of index and let $b_{n}(G)$ be the number of normal subgroups of index . We show that $a_{p^{k}}(SL_{2}^{1}(\mathbb{F}_{p}t]])) \le p^{k(k+5)/2}$ for . If $\Lambda=\mathbb{F}_{p}t]]$ and does not divide or if $\Lambda=\mathbb{Z}_{p}$ and $p \ne 2$ or $d \ne 2$ , we show that for all sufficiently large $b_{p^{k}}(SL_{d}^{1}(\Lambda))=b_{p^{k+d^{2}-1}}(SL_{d}^{1}(\Lambda))$ . On the other hand if $\Lambda=\mathbb{F}_{p}t]]$ and divides , then $b_{n}(SL_{d}^{1}(\Lambda))$ is not even bounded as a function of .

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