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 Baumslag–Solitar 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 Baumslag–Solitar 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