  We consider Turing machines as actions over configurations in
$\Sigma^{\mathbb{Z}^d}$ which only change them locally around a marked position
that can move and carry a particular state. In this setting we study the monoid
of Turing machines and the group of reversible Turing machines. We also study
two natural subgroups, namely the group of finite-state automata, which
generalizes the topological full groups studied in the theory of
orbit-equivalence, and the group of oblivious Turing machines whose movement is
independent of tape contents, which generalizes lamplighter groups and has
connections to the study of universal reversible logical gates. Our main
results are that the group of Turing machines in one dimension is neither
amenable nor residually finite, but is locally embeddable in finite groups, and
that the torsion problem is decidable for finite-state automata in dimension
one, but not in dimension two.
