Tóm tắt
Abstract
We explore the topological full group
〚
G
〛
$\llbracket G\rrbracket $
of
an essentially principal étale groupoid G on a Cantor set.
When G is minimal,
we show that
〚
G
〛
$\llbracket G\rrbracket $
(and its certain normal subgroup) is a complete invariant
for the isomorphism class of the étale groupoid G.
Furthermore, when G is either almost finite or purely infinite,
the commutator subgroup
D
(
〚
G
〛
)
$D(\llbracket G\rrbracket )$
is shown to be simple.
The étale groupoid G
arising from a one-sided irreducible shift of finite type is
a typical example of a purely infinite minimal groupoid.
For such G,
〚
G
〛
$\llbracket G\rrbracket $
is thought of as a generalization of the Higman–Thompson group.
We prove that
〚
G
〛
$\llbracket G\rrbracket $
is of type F
∞,
and so in particular it is finitely presented.
This gives us a new infinite family of
finitely presented infinite simple groups.
Also, the abelianization of
〚
G
〛
$\llbracket G\rrbracket $
is calculated and described
in terms of the homology groups of G.
