  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 $\mathbf{Z}$. The second ensures that irreducible
cocompact lattices in a product of non-discrete 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 non-discrete Burger-Mozes universal group of
automorphisms of a tree with an arbitrary prescribed local action admits a
compactly generated closed subgroup with a non-discrete simple quotient. As
applications, we answer a question of D. Wise by proving the non-residual
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.
