58 research outputs found
A geometric approach to (semi)-groups defined by automata via dual transducers
We give a geometric approach to groups defined by automata via the notion of
enriched dual of an inverse transducer. Using this geometric correspondence we
first provide some finiteness results, then we consider groups generated by the
dual of Cayley type of machines. Lastly, we address the problem of the study of
the action of these groups in the boundary. We show that examples of groups
having essentially free actions without critical points lie in the class of
groups defined by the transducers whose enriched dual generate a torsion-free
semigroup. Finally, we provide necessary and sufficient conditions to have
finite Schreier graphs on the boundary yielding to the decidability of the
algorithmic problem of checking the existence of Schreier graphs on the
boundary whose cardinalities are upper bounded by some fixed integer
Generalizations of the Muller-Schupp theorem and tree-like inverse graphs
We extend the characterization of context-free groups of Muller and Schupp in
two ways. We first show that for a quasi-transitive inverse graph ,
being quasi-isometric to a tree, or context-free (finitely many end-cones
types), or having the automorphism group that is virtually free,
are all equivalent conditions. Furthermore, we add to the previous equivalences
a group theoretic analog to the representation theorem of
Chomsky-Sch\"utzenberger that is fundamental in solving a weaker version of a
conjecture of T. Brough which also extends Muller and Schupp' result to the
class of groups that are virtually finitely generated subgroups of direct
product of free groups. We show that such groups are precisely those whose word
problem is the intersection of a finite number of languages accepted by
quasi-transitive, tree-like inverse graphs
Groups and Semigroups Defined by Colorings of Synchronizing Automata
In this paper we combine the algebraic properties of Mealy machines
generating self-similar groups and the combinatorial properties of the
corresponding deterministic finite automata (DFA). In particular, we relate
bounded automata to finitely generated synchronizing automata and characterize
finite automata groups in terms of nilpotency of the corresponding DFA.
Moreover, we present a decidable sufficient condition to have free semigroups
in an automaton group. A series of examples and applications is widely
discussed, in particular we show a way to color the De Bruijn automata into
Mealy automata whose associated semigroups are free, and we present some
structural results related to the associated groups
On periodic points of free inverse monoid endomorphisms
It is proved that the periodic point submonoid of a free inverse monoid
endomorphism is always finitely generated. Using Chomsky's hierarchy of
languages, we prove that the fixed point submonoid of an endomorphism of a free
inverse monoid can be represented by a context-sensitive language but, in
general, it cannot be represented by a context-free language.Comment: 18 page
Freeness of automaton groups vs boundary dynamics
We prove that the boundary dynamics of the (semi)group generated by the enriched dual transducer characterizes the algebraic property of being free for an automaton group. We specialize this result to the class of bireversible transducers and we show that the property of being not free is equivalent to the existence of a finite Schreier graph in the boundary of the enriched dual pointed at some essentially non-trivial point. From these results we derive some consequences from the algebraic, algorithmic and dynamical points of view
Fixed points of endomorphisms of graph groups
It is shown, for a given graph group , that the fixed point subgroup
Fix is finitely generated for every endomorphism of if
and only if is a free product of free abelian groups. The same conditions
hold for the subgroup of periodic points. Similar results are obtained for
automorphisms, if the dependence graph of is a transitive forest.Comment: 9 page
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