On the commensurator of the Nottingham group

Ershov, Mikhail

doi:10.1090/S0002-9947-2010-05160-8

On the commensurator of the Nottingham group
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Trans. Amer. Math. Soc. 362 (2010), 6663-6678 Request permission

Abstract:

Let $p\geq 5$ be a prime number. We prove that the abstract commensurator of the Nottingham group $\mathcal {N}(\mathbb {F}_p)$ coincides with its automorphism group, which is known to be a finite extension of $\mathcal {N}(\mathbb {F}_p)$. As a corollary we deduce that the Nottingham group cannot be embedded as an open subgroup of a topologically simple group.

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
Miklós Abért and Bálint Virág, Dimension and randomness in groups acting on rooted trees, J. Amer. Math. Soc. 18 (2005), no. 1, 157–192. MR 2114819, DOI 10.1090/S0894-0347-04-00467-9
Y. Barnea, M. Ershov and T. Weigel, Abstract commensurators of profinite groups, to appear in Transactions of the AMS.
Y. Barnea, N. Gavioli, A. Jaikin-Zapirain, V. Monti, and C. M. Scoppola, Pro-$p$ groups with few normal subgroups, J. Algebra 321 (2009), no. 2, 429–449. MR 2483275, DOI 10.1016/j.jalgebra.2008.10.012
Yiftach Barnea and Benjamin Klopsch, Index-subgroups of the Nottingham group, Adv. Math. 180 (2003), no. 1, 187–221. MR 2019222, DOI 10.1016/S0001-8708(02)00102-0
Yiftach Barnea and Aner Shalev, Hausdorff dimension, pro-$p$ groups, and Kac-Moody algebras, Trans. Amer. Math. Soc. 349 (1997), no. 12, 5073–5091. MR 1422889, DOI 10.1090/S0002-9947-97-01918-1
Y. Barnea, A. Shalev, and E. I. Zelmanov, Graded subalgebras of affine Kac-Moody algebras, Israel J. Math. 104 (1998), 321–334. MR 1622319, DOI 10.1007/BF02897069
Rachel Camina, The Nottingham group, New horizons in pro-$p$ groups, Progr. Math., vol. 184, Birkhäuser Boston, Boston, MA, 2000, pp. 205–221. MR 1765121
Mikhail Ershov, New just-infinite pro-$p$ groups of finite width and subgroups of the Nottingham group, J. Algebra 275 (2004), no. 1, 419–449. MR 2047455, DOI 10.1016/j.jalgebra.2003.08.012
Benjamin Klopsch, Automorphisms of the Nottingham group, J. Algebra 223 (2000), no. 1, 37–56. MR 1738250, DOI 10.1006/jabr.1999.8040
Richard Pink, Compact subgroups of linear algebraic groups, J. Algebra 206 (1998), no. 2, 438–504. MR 1637068, DOI 10.1006/jabr.1998.7439