  We revisit our construction of the Thompson groups from the polycyclic
inverse monoids in the light of new research. Specifically, we prove that the
Thompson group $G_{n,1}$ is the group of units of a Boolean inverse monoid
$C_{n}$ called the Cuntz inverse monoid. This inverse monoid is proved to be
the tight completion of the polycyclic inverse monoid $P_{n}$. The \'etale
topological groupoid associated with $C_{n}$ under non-commutative Stone
duality is the usual groupoid associated with the corresponding Cuntz
$C^{\ast}$-algebra. We then show that the group $G_{n,1}$ is also the group of
automorphisms of a specific $n$-ary Cantor algebra: this $n$-ary Cantor algebra
is constructed first as the monoid of total maps of a restriction semigroup \`a
la Statman and then in terms of labelled trees \`a la Higman.
