6 research outputs found
Submonoids and rational subsets of groups with infinitely many ends
In this paper we show that the membership problems for finitely generated
submonoids and for rational subsets are recursively equivalent for groups with
two or more ends
Knapsack Problems in Groups
We generalize the classical knapsack and subset sum problems to arbitrary
groups and study the computational complexity of these new problems. We show
that these problems, as well as the bounded submonoid membership problem, are
P-time decidable in hyperbolic groups and give various examples of finitely
presented groups where the subset sum problem is NP-complete.Comment: 28 pages, 12 figure
Undecidability of the word problem for one-relator inverse monoids via right-angled Artin subgroups of one-relator groups
We prove the following results: (1) There is a one-relator inverse monoid Inv⟨A|w=1⟩ with undecidable word problem; and (2) There are one-relator groups with undecidable submonoid membership problem. The second of these results is proved by showing that for any finite forest the associated right-angled Artin group embeds into a one-relator group. Combining this with a result of Lohrey and Steinberg (J Algebra 320(2):728–755, 2008), we use this to prove that there is a one-relator group containing a fixed finitely generated submonoid in which the membership problem is undecidable. To prove (1) a new construction is introduced which uses the one-relator group and submonoid in which membership is undecidable from (2) to construct a one-relator inverse monoid Inv⟨A|w=1⟩ with undecidable word problem. Furthermore, this method allows the construction of an E-unitary one-relator inverse monoid of this form with undecidable word problem. The results in this paper answer a problem originally posed by Margolis et al. (in: Semigroups and their applications, Reidel, Dordrecht, pp. 99–110, 1987)
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic