Hausdorff dimension, pro-𝑝 groups, and Kac-Moody algebras

Barnea, Yiftach; Shalev, Aner

doi:10.1090/S0002-9947-97-01918-1

Hausdorff dimension, pro-$p$ groups, and Kac-Moody algebras
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by Yiftach Barnea and Aner Shalev PDF

Trans. Amer. Math. Soc. 349 (1997), 5073-5091 Request permission

Abstract:

Every finitely generated profinite group can be given the structure of a metric space, and as such it has a well defined Hausdorff dimension function. In this paper we study Hausdorff dimension of closed subgroups of finitely generated pro-$p$ groups $G$. We prove that if $G$ is $p$-adic analytic and $H \le _c G$ is a closed subgroup, then the Hausdorff dimension of $H$ is $\dim H/\dim G$ (where the dimensions are of $H$ and $G$ as Lie groups). Letting the spectrum $\operatorname {Spec}(G)$ of $G$ denote the set of Hausdorff dimensions of closed subgroups of $G$, it follows that the spectrum of $p$-adic analytic groups is finite, and consists of rational numbers. We then consider some non-$p$-adic analytic groups $G$, and study their spectrum. In particular we investigate the maximal Hausdorff dimension of non-open subgroups of $G$, and show that it is equal to $1 - {1 \over {d+1}}$ in the case of $G = SL_d(F_p[[t]])$ (where $p > 2$), and to $1/2$ if $G$ is the so called Nottingham group (where $p >5$). We also determine the spectrum of $SL_2(F_p[[t]])$ ($p>2$) completely, showing that it is equal to $[0,2/3] \cup \{ 1 \}$. Some of the proofs rely on the description of maximal graded subalgebras of Kac-Moody algebras, recently obtained by the authors in joint work with E. I. Zelmanov.

References

A. G. Abercrombie, Subgroups and subrings of profinite rings, Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 2, 209–222. MR 1281541, DOI 10.1017/S0305004100072522
N. Bourbaki, Lie groups and Lie algebras, Chapters 1-3, Springer, Berlin, 1980.
Y. Barnea, A. Shalev and E.I. Zelmanov, Graded subalgebras of affine Kac-Moody algebras, to appear in Israel J. Math.
R. Camina, Subgroups of the Nottingham Group, Ph.D. Thesis, QMW, London, 1996.
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-$p$-groups, London Mathematical Society Lecture Note Series, vol. 157, Cambridge University Press, Cambridge, 1991. MR 1152800
Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842, DOI 10.1007/978-1-4612-6398-2
D. L. Johnson, The group of formal power series under substitution, J. Austral. Math. Soc. Ser. A 45 (1988), no. 3, 296–302. MR 957195, DOI 10.1017/S1446788700031001
Michel Lazard, Groupes analytiques $p$-adiques, Inst. Hautes Études Sci. Publ. Math. 26 (1965), 389–603 (French). MR 209286
Alexander Lubotzky and Avinoam Mann, Powerful $p$-groups. I. Finite groups, J. Algebra 105 (1987), no. 2, 484–505. MR 873681, DOI 10.1016/0021-8693(87)90211-0
Alexander Lubotzky and Aner Shalev, On some $\Lambda$-analytic pro-$p$ groups, Israel J. Math. 85 (1994), no. 1-3, 307–337. MR 1264349, DOI 10.1007/BF02758646
Jean-Pierre Serre, Lie algebras and Lie groups, 2nd ed., Lecture Notes in Mathematics, vol. 1500, Springer-Verlag, Berlin, 1992. 1964 lectures given at Harvard University. MR 1176100, DOI 10.1007/978-3-540-70634-2
Aner Shalev, Growth functions, $p$-adic analytic groups, and groups of finite coclass, J. London Math. Soc. (2) 46 (1992), no. 1, 111–122. MR 1180887, DOI 10.1112/jlms/s2-46.1.111
Aner Shalev, Some problems and results in the theory of pro-$p$ groups, Groups ’93 Galway/St. Andrews, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 212, Cambridge Univ. Press, Cambridge, 1995, pp. 528–542. MR 1337295, DOI 10.1017/CBO9780511629297.019
I.O. York, The Group of Formal Power Series under Substitution, Ph.D. Thesis, Nottingham, 1990.
E. I. Zel′manov, On periodic compact groups, Israel J. Math. 77 (1992), no. 1-2, 83–95. MR 1194787, DOI 10.1007/BF02808012