From some Pisot numerations to topological groups

A Pisot numeration system $U$ for $\mathbb N$ is a sequence of natural numbers
generated by an integral homogeneous linear recurrence whose
characteristic polynomial is the minimal polynomial of a Pisot number.
The purpose of this paper is to introduce the analogue of the group of
$p$-adic integers for such numerations when they \emph{preserve zeros},
which is equivalent to the `Condition F' introduced by Frougny and
Solomyak for $\beta$-numerations. We show that these topological groups $\mathbb Z_U$
project homomorphically onto a torus. Equipping $\mathbb Z_U$ with the
appropriate topology, we also show that if $U$ is unimodular, then $\mathbb Z_U$
is continuously isomorphic to a torus.