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## Print version ISSN 0034-7426

### Rev.colomb.mat. vol.39 no.1 Bogotá Jan./June 2005

On certain closed subgroups of SL (2, Zp[[X]])

Álvaro Lozano-Robledo

Dept. of Mathematics. Colby College, Waterville Maine 04901, USA.

e-mail: alozano@colby.edu

Abstract. Let p > 2 be a prime number and let Λ = Zp[[X]] be the ring of power series with p-adic integer coefficients. The special linear group of matrices SL(2, Λ) is equipped with several natural projections. In particular, let πX: SL(2, Λ) → SL(2; Zp) be the natural projection which sends X → 0. Suppose that G is a subgroup of SL(2; Λ) such that the projection H = πX(G) is known. In this note, different criteria are found which guarantee that the subgroup G of SL(2; Λ) is "as large as possible", i.e. G is the full inverse image of H. Criteria of this sort have interesting applications in the theory of Galois representations.

Keywords and phrases. Closed subgroups, special linear group, Iwasawa algebra.

2000 Mathematics Subject Classiffication. Primary: 15A33, 15A54, Secondary: 11F80.

Resumen. Sea p > 2 un primo y Λ = Zp[[X]] el anillo de series de potencias con coefficientes enteros p-adicos. El grupo lineal de matrices especial SL(2, Λ) es equipado con varias proyecciones naturales. En particular, πX: SL(2, Λ) → SL(2, Zp) es la proyección natural que envia X → 0. Suponga que G es un subgrupo de SL(2, Λ) tal que la proyección H = πX(G) es conocida. En este artículo se establecen diferentes criterios que garantizan que el subgrupo G de SL(2, Λ) es "tan grande como es posible"; esto es, G es la imagen inversa total de H. Criterios de esta naturaleza tienen importantes aplicaciones a la teoría de representaciones de Galois.

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References

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(Recibido en abril de 2005. Aceptado en mayo de 2005)