Claas E. Röver

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 co-word 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.

Groups with indexed co-word problem

We investigate co-indexed groups, that is groups whose co-word problem (all words defining nontrivial elements) is an indexed language. We show that all Higman–Thompson groups and a large class of tree automorphism groups defined by finite automata are co-indexed groups. The latter class is closely related to dynamical systems and includes the Grigorchuk 2-group and the Gupta–Sidki 3-group. The co-word problems of all these examples are in fact accepted by nested stack automata with certain additional properties, and we establish various closure properties of this restricted class of co-indexed groups, including closure under free products.

Abstract Commensurators of Groups Acting on Rooted Trees

AbstractWe show that the abstract commensurator of a nearly level transitive weakly branch group H coincides with the relative commensurator of H in the homeomorphism group of the boundary of the tree on which H acts. It is also shown that the commensurator of an infinite group which is commensurable with its own nth direct power

Commensurations and metric properties of Houghton’s groups

(2 \leqslant n \in \mathbb{N})

GROUPS WITH CONTEXT-FREE CONJUGACY PROBLEMS

contains a Higman–Thompson group as a subgroup. Applying these results to ‘the’ Grigorchuk 2-group G we show that the commensurator of G is a finitely presented infinite simple group.

On real-time word problems

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. We show Com.F/ is not finitely generated and determine which subgroups of finite index in F are isomorphic to F . We also show that the natural map from the commensurator group to the quasi-isometry group of F is injective. 20E34, 20F65; 26A30, 20F28

ON GROUPS WHICH ARE SYNTACTIC MONOIDS OF DETERMINISTIC CONTEXT-FREE LANGUAGES

We describe the automorphism groups and the abstract commensurators of Houghtons 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.

Addendum to “Commensurations and subgroups of finite index of Thompson’s group F”

Abstract The normalizer NW (WJ ) of a standard parabolic subgroup WJ of a finite Coxeter group W splits over the parabolic subgroup with complement NJ consisting of certain minimal length coset representatives of WJ 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 Dn ) the centralizer CW (w) of an element w ∈ W is in a similar way a semidirect product of the centralizer of w in a suitable small parabolic subgroup WJ with complement isomorphic to the normalizer complement NJ . Then we use this result to give a new short proof of Solomons Character Formula and discuss its connection to MacMahons master theorem.