40 research outputs found
Groups with context-free co-word problem
The class of co-context-free groups is studied. A co-context-free group is defined as one whose coword
problem (the complement of its word problem) is context-free. This class is larger than the
subclass of context-free groups, being closed under the taking of finite direct products, restricted
standard wreath products with context-free top groups, and passing to finitely generated subgroups
and finite index overgroups. No other examples of co-context-free groups are known. It is proved
that the only examples amongst polycyclic groups or the BaumslagâSolitar groups are virtually
abelian. This is done by proving that languages with certain purely arithmetical properties cannot
be context-free; this result may be of independent interest
Groups and semigroups with a one-counter word problem
We prove that a finitely generated semigroup whose word problem is a one-counter language has a linear growth function. This provides us with a very strong restriction on the structure of such a semigroup, which, in particular, yields an elementary proof of a result of Herbst, that a group with a one-counter word problem is virtually cyclic. We prove also that the word problem of a group is an intersection of finitely many one-counter languages if and only if the group is virtually abelian
On Cayley graphs of virtually free groups
In 1985, Dunwoody showed that finitely presentable groups are accessible.
Dunwoody's result was used to show that context-free groups, groups
quasi-isometric to trees or finitely presentable groups of asymptotic dimension
1 are virtually free. Using another theorem of Dunwoody of 1979, we study when
a group is virtually free in terms of its Cayley graph and we obtain new proofs
of the mentioned results and other previously depending on them
Context-free pairs of groups I: Context-free pairs and graphs
Let be a finitely generated group, a finite set of generators and
a subgroup of . We call the pair context-free if the set of all
words over that reduce in to an element of is a context-free
language. When is trivial, itself is called context-free; context-free
groups have been classified more than 20 years ago in celebrated work of Muller
and Schupp as the virtually free groups.
Here, we derive some basic properties of such group pairs. Context-freeness
is independent of the choice of the generating set. It is preserved under
finite index modifications of and finite index enlargements of . If
is virtually free and is finitely generated then is context-free. A
basic tool is the following: is context-free if and only if the
Schreier graph of with respect to is a context-free graph