3 Lattices for products of trees
Let
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:be locally finite trees. Then the group
dvipng error! exitcode was 2 (signal 0), transscript follows:is isomorphic to
dvipng error! exitcode was 2 (signal 0), transscript follows:. A lattice
dvipng error! exitcode was 2 (signal 0), transscript follows:is irreducible if it is not commensurable to a product
dvipng error! exitcode was 2 (signal 0), transscript follows:of lattices
dvipng error! exitcode was 2 (signal 0), transscript follows:for
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Problem 3.1 (Comments) [Burger--Mozes] Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be an irreducible uniform lattice in
dvipng error! exitcode was 2 (signal 0), transscript follows:. For
dvipng error! exitcode was 2 (signal 0), transscript follows:, let
dvipng error! exitcode was 2 (signal 0), transscript follows:be the closure of the projection
dvipng error! exitcode was 2 (signal 0), transscript follows:. What are the possible
dvipng error! exitcode was 2 (signal 0), transscript follows:s? What about nonuniform lattices?
Remark We assume the projections are locally primitive, that is, that the restriction of their action to the star of any vertex of the tree
dvipng error! exitcode was 2 (signal 0), transscript follows:is primitive. If the irreducible lattice
dvipng error! exitcode was 2 (signal 0), transscript follows:is uniform then the projections cannot be dense in either of
dvipng error! exitcode was 2 (signal 0), transscript follows:(Burger--Mozes MR1839489). If
dvipng error! exitcode was 2 (signal 0), transscript follows:has an infinite linear representation over characteristic
dvipng error! exitcode was 2 (signal 0), transscript follows:, then
dvipng error! exitcode was 2 (signal 0), transscript follows:is an extension of an arithmetic lattice (Burger--Mozes--Zimmer BurgerMozesZimmer). D. Rattaggi's thesis Rattaggi gives many interesting examples.
Problem 3.2 (Comments) [Burger--Mozes] Study lattices in products of three or more trees.
Remark One would like to construct interesting lattices. Here "interesting" means, in particular, examples which are not arithmetic or which are not extensions of arithmetic lattices. It is not obvious how to make deformations with
dvipng error! exitcode was 2 (signal 0), transscript follows:trees. Maybe these examples will give new phenomena e.g. a group which is residually finite and not linear?
Problem 3.3 (Comments) [Burger--Mozes] Define and study lattices in an infinite product of trees: "adeles".
Remark There is a chance that all lattices are arithmetic here.