summary：

A tree-theoretic approach to classify lattices in PGL_2 of

p-adic fields, developed in my past works, partly by collaboration

with G.Cornelissen and A.Kontogeorgis, gave an effective way to

describe lattices in this p-adic Lie group, and was applied to

numerous problems in geometry of algebraic curves. Daniel Allcock and

I tried to carry out the similar story for PGL_3. What we obtained so

far are the following results, which I am going to speak about:

PGL_3(Q_2) has exactly two lattices of minimal covolume, which are

both arithmetic, commensurable to each other; moreover, one of these

two lattices is the one constructed by Mumford in his famous

construction of a fake projective plane.