 
Summary: THE ARBOREAL APPROACH
TO PAIRS OF INVOLUTIONS IN RANK TWO
JAUME AGUAD´E
1. Introduction
We consider representations of the infinite dihedral group D in
GL2(Zp) (Zp is the ring of padic integers for a chosen prime p). Each
of these representations is given by a pair of involutions 1, 2 up to
conjugation. These representations were classified in [1] using some nu
merical invariants which were introduced in that paper in a completely
formal way. Actually, these invariants appeared in a natural way in
the computations of the mod p cohomology of the classifying spaces
of rank two KacMoody groups and some related spaces as discussed
in [2], but the proofs in [1] are independent of all the topological ma
chinery in [2]. The interested reader may read section 7 in [1] for a
quick overview of the relationship between representations of D and
rank two KacMoody groups.
In the present paper we provide a new classification of the represen
tations of D in GL2(Zp) and new proofs for the classification the
orems in [1]. The proofs that we present here are simpler and more
illuminating than the proofs in [1]. These new proofs are geometrical,
