  Much work has gone into matrix representations of Galois groups, but there is
a whole new class of naturally occurring representations that have as yet gone
almost unnoticed. In fact, it is well-known in various areas of mathematics
that the main sources of totally disconnected groups are matrix groups over
local fields AND automorphism groups of locally finite trees. It is perhaps
surprising then that representations of Galois groups into the latter have been
almost ignored, while at the same time Galois representations into the former
have been enormously effective in resolving long-standing problems in number
theory.
  These ``tree'' representations are important as regards topics such as the
unramified Fontaine-Mazur conjecture. This conjecture states that any p-adic
representation of the Galois group of an extension unramified at p (and
ramified at only finitely many primes) should have finite image. In other
words, p-adic representations say little about such Galois groups. This paper
proposes the conjecture that these Galois groups should, on the other hand,
have representations with large image (measured by Hausdorff dimension) in the
automorphism group of a rooted tree. Thus, they might allow us to investigate
the structure of the Galois group of infinite pro-p extensions (such as Hilbert
p-class towers), something unapproachable by standard p-adic representation
methods.
