We establish three independent results on groups acting on trees. The first implies
that a compactly generated locally compact group which acts continuously on a
locally finite tree with nilpotent local action and no global fixed point is virtually
indicable; that is to say, it has a finite-index subgroup which surjects onto
. The
second ensures that irreducible cocompact lattices in a product of nondiscrete locally
compact groups such that one of the factors acts vertex-transitively on a tree with a
nilpotent local action cannot be residually finite. This is derived from a general
result, of independent interest, on irreducible lattices in product groups. The third
implies that every nondiscrete Burger–Mozes universal group of automorphisms of a
tree with an arbitrary prescribed local action admits a compactly generated
closed subgroup with a nondiscrete simple quotient. As applications, we
answer a question of D Wise by proving the nonresidual finiteness of a certain
lattice in a product of two regular trees, and we obtain a negative answer to
a question of C Reid, concerning the structure theory of locally compact
groups.
Keywords
trees, lattices in products, locally compact groups
