 
Summary: THE DENSEST LATTICES IN PGL3(Q2)
DANIEL ALLCOCK AND FUMIHARU KATO
Abstract. We find the smallest possible covolume for lattices
in PGL3(Q2), show that there are exactly two lattices with this
covolume, and describe them explicitly. They are commensurable,
and one of them appeared in Mumford's construction of his fake
projective plane.
The most famous lattice in the projective group PGL3(Q2) over the
2adic rational numbers Q2 is the one Mumford used to construct his
fake projective plane [22]. Namely, he found an arithmetic group P1
(we call it PM ) containing a torsionfree subgroup of index 21, such
that the algebraic surface associated to it by the theory of padic uni
formization [23, 24] is a fake projective plane. The full classification of
fake projective planes has been obtained recently [26].
The second author and his collaborators have developed a diagram
matic calculus [8, 15] for working with algebraic curves (including orb
ifolds) arising from padic uniformization using lattices in PGL2 over
a nonarchimedean local field. It allows one to read off properties of
the curves from the quotient of the BruhatTits tree and to construct
lattices with various properties, or prove they don't exist. We hope
