16 research outputs found
On real-time word problems
It is proved that the word problem of the direct product of two free groups of rank 2 can be recognised by a 2-tape real-time but not by a 1-tape real-time Turing machine. It is also proved that the BaumslagâSolitar groups B(1,r) have the 5-tape real-time word problem for all r != 0
Commensurations and Metric Properties of Houghton's Groups
We describe the automorphism groups and the abstract commensurators of
Houghton's groups. Then we give sharp estimates for the word metric of these
groups and deduce that the commensurators embed into the corresponding
quasi-isometry groups. As a further consequence, we obtain that the Houghton
group on two rays is at least quadratically distorted in those with three or
more rays
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
Commensurations and Subgroups of Finite Index of Thompson's Group F
We determine the abstract commensurator com(F) of Thompson's group F and
describe it in terms of piecewise linear homeomorphisms of the real line and in
terms of tree pair diagrams. We show com (F) is not finitely generated and
determine which subgroups of finite index in F are isomorphic to F. We show
that the natural map from the commensurator group to the quasi-isometry group
of F is injective.Comment: 9 page
On the distortion of twin building lattices
We show that twin building lattices are undistorted in their ambient group;
equivalently, the orbit map of the lattice to the product of the associated
twin buildings is a quasi-isometric embedding. As a consequence, we provide an
estimate of the quasi-flat rank of these lattices, which implies that there are
infinitely many quasi-isometry classes of finitely presented simple groups. In
an appendix, we describe how non-distortion of lattices is related to the
integrability of the structural cocycle
Constructing finitely presented simple groups that contain Grigorchuk groups
We construct infinite finitely presented simple groups that have subgroups isomorphic to Grigorchuk groups. We also prove that up to one possible exception all previously known finitely presented simple groups are torsion locally finite.</p
A note on element centralizers in finite Coxeter groups
The normalizer N-W(W-J) of a standard parabolic subgroup W-J of a finite Coxeter group W splits over the parabolic subgroup with complement N-J consisting of certain minimal length coset representatives of W-J in W. In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type D-n) the centralizer C-W(w) of an element w epsilon W is in a similar way a semidirect product of the centralizer of w in a suitable small parabolic subgroup W-J with complement isomorphic to the normalizer complement N-J. Then we use this result to give a new short proof of Solomon's Character Formula and discuss its connection to MacMahon's master theorem.The second author wishes to acknowledge support from Science
Foundation Ireland.peer-reviewe