10 research outputs found
Groups, Graphs, Languages, Automata, Games and Second-order Monadic Logic
In this paper we survey some surprising connections between group theory, the
theory of automata and formal languages, the theory of ends, infinite games of
perfect information, and monadic second-order logic
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
Context-freeness of the languages of Schützenberger automata of HNN-extensions of finite inverse semigroups
We prove that the Schützenberger graph of any element of the HNN-extension of a finite inverse semigroup S with respect to its standard presentation is a context-free graph in the sense of [11], showing that the language L recognized by this automaton is context-free. Finally we explicitly construct the grammar generating L, and from this fact we show that the word problem for an HNN-extension of a finite inverse semigroup S is decidable and lies in the complexity class of polynomial time problems
Logspace computations in graph products
We consider three important and well-studied algorithmic problems in group
theory: the word, geodesic, and conjugacy problem. We show transfer results
from individual groups to graph products. We concentrate on logspace complexity
because the challenge is actually in small complexity classes, only. The most
difficult transfer result is for the conjugacy problem. We have a general
result for graph products, but even in the special case of a graph group the
result is new. Graph groups are closely linked to the theory of Mazurkiewicz
traces which form an algebraic model for concurrent processes. Our proofs are
combinatorial and based on well-known concepts in trace theory. We also use
rewriting techniques over traces. For the group-theoretical part we apply
Bass-Serre theory. But as we need explicit formulae and as we design concrete
algorithms all our group-theoretical calculations are completely explicit and
accessible to non-specialists
Subgroups of direct products of graphs of groups with free abelian vertex groups
A result of Baumslag and Roseblade states that a finitely presented subgroup
of the direct product of two free groups is virtually a direct product of free
groups. In this paper we generalise this result to the class of cyclic subgroup
separable graphs of groups with free abelian vertex groups and cyclic edge
groups. More precisely, we show that a finitely presented subgroup of the
direct product of two groups in this class is virtually -by-(free abelian),
where is the direct product of two groups in the class. In particular, our
result applies to 2-dimensional coherent right-angled Artin groups and
residually finite tubular groups. Furthermore, we show that the multiple
conjugacy problem and the membership problem are decidable for finitely
presented subgroups of the direct product of two -dimensional coherent
RAAGs
Solutions of Word Equations over Partially Commutative Structures
We give NSPACE(n log n) algorithms solving the following decision problems.
Satisfiability: Is the given equation over a free partially commutative monoid
with involution (resp. a free partially commutative group) solvable?
Finiteness: Are there only finitely many solutions of such an equation? PSPACE
algorithms with worse complexities for the first problem are known, but so far,
a PSPACE algorithm for the second problem was out of reach. Our results are
much stronger: Given such an equation, its solutions form an EDT0L language
effectively representable in NSPACE(n log n). In particular, we give an
effective description of the set of all solutions for equations with
constraints in free partially commutative monoids and groups