15 research outputs found
Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups
We study a class M of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defining relations. This class includes many finite metacyclic generalized Fibonacci groups that have been previously identified in the literature. By analysing their shift extensions we show that the groups in the class M are are coherent, sub-group separable, satisfy the Tits alternative, possess finite index subgroups of geometric dimension at most two, and that their finite subgroups are all meta-cyclic. Many of the groups in M are virtually free, some are free products of metacyclic groups and free groups, and some have geometric dimension two. We classify the finite groups that occur in M, giving extensive details about the metacyclic structures that occur, and we use this to prove an earlier conjecture concerning cyclically presented groups in which the relators are positive words of length three. We show that any finite group in the class M that has fixed point free shift automorphism must be cyclic
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Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups
We study a class M of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defining relations. This class includes many finite metacyclic generalized Fibonacci groups that have been previously identified in the literature. By analyzing their shift extensions we show that the groups in the class M are coherent, subgroup separable, satisfy the Tits alternative, possess finite index subgroups of geometric dimension at most two, and that their finite subgroups are all metacyclic. Many of the groups in are virtually free, some are free products of metacyclic groups and free groups, and some have geometric dimension two. We classify the finite groups that occur in , giving extensive details about the metacyclic structures that occur, and we use this to prove an earlier conjecture concerning cyclically presented groups in which the relators are positive words of length three. We show that any finite group in the class M that has fixed point free shift automorphism must be cyclic
The language of self-avoiding walks
Let be an infinite, locally finite, connected graph without
loops or multiple edges. We consider the edges to be oriented, and is
equipped with an involution which inverts the orientation. Each oriented edge
is labelled by an element of a finite alphabet . The labelling
is assumed to be deterministic: edges with the same initial (resp. terminal)
vertex have distinct labels. Furthermore it is assumed that the group of
label-preserving automorphisms of acts quasi-transitively. For any vertex
of , consider the language of all words over which can
be read along self-avoiding walks starting at . We characterize under which
conditions on the graph structure this language is regular or context-free.
This is the case if and only if the graph has more than one end, and the size
of all ends is , or at most , respectively.Comment: 24 pages, 3 figure
Unsolved Problems in Group Theory. The Kourovka Notebook
This is a collection of open problems in group theory proposed by hundreds of
mathematicians from all over the world. It has been published every 2-4 years
in Novosibirsk since 1965. This is the 19th edition, which contains 111 new
problems and a number of comments on about 1000 problems from the previous
editions.Comment: A few new solutions and references have been added or update
IDATER online conference: graphicacy and modelling 2010
IDATER online conference: graphicacy and modelling 201