The Arboreal Approach to Pairs of Involutions in Rank Two

Abstract

We use ideas of J.-P. Serre to obtain a geometric classification of the integral p-adic rank two representations of the infinite dihedral group.

Key Words:

• Infinite dihedral group
• Trees

2000 Mathematics Subject Classification:

• 20C12
• 5C05

ACKNOWLEDGMENTS

It is worthwhile to explain here how was the genesis of this article. The author was fortunate enough to have Jean-Pierre Serre in the audience of a talk on the work in Aguadé et al. (2007) that he gave at a meeting of the Tunisian Mathematical Society. Serre immediately suggested that the abstract numerical invariants of Aguadé et al. (2007) should have a natural geometrical interpretation in terms of the tree of GL ₂(ℤ _p ), and he outlined how this interpretation should look like. Hence, the present article can be viewed as a writing out of Serre's ideas together with some results needed to understand the relationship between the classification in Aguadé et al. (2007) and the classification that we obtain here. The author is in deep gratitude to J.-P. Serre for sharing his ideas with him and would also like to thank S. Zarati for his invitation to Tunis where this work began. The hospitality of PIMS in Vancouver is also gratefully acknowledged.

The author is partially supported by grants MTM204-06686, SGR2005-00606, and PR2007-0097.

Notes

Communicated by M. Vigué.