11 research outputs found
Definability of maximal cofinitary groups
We present a proof of a result, previously announced by the second author,
that there is a closed (even ) set generating an (even
) maximal cofinitary group (short, mcg) which is isomorphic to a
free group. In this isomorphism class, this is the lowest possible definitional
complexity of an mcg.Comment: This work is part of the first authors thesi
Infinite Permutation Groups
This is a transcript of a lecture course on Infinite Permutation Groups given
by Peter M. Neumann (1940-2020) in Oxford during the academic year 1988-1989.
The field of Infinite Permutation Groups only emerged as an independent field
of study in the 1980's. Most of the results described in these notes were at
the time of the lectures brand new and had either just recently appeared in
print or had not appeared formally. A large part of the results described is
either due to Peter himself or heavily influenced by him. These notes offer
Peter's personal take on a field that he was instrumental in creating and in
many cases ideas and questions that can not be found in the published
literature.Comment: The text of [v1] and [v2] is identical. [v1] is formatted so that
each new lecture starts at a right hand page (127+v pages) and [v2] is
formatted without all the extra page breaks (93+v pages). Edited by David A.
Craven (Birmingham), Dugald Macpherson (Leeds) and R\"ognvaldur G. M\"oller
(Reykjav\'ik). Comments and corrections can be sent to R\"oggi M\"oller
([email protected]
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
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
Combinatorics of countable ordinal topologies
We study combinatorial properties of ordinals under the order topology, focusing on the subspaces, partition properties and autohomeomorphism groups of countable ordinals. Our main results concern topological partition relations. Let n be a positive integer, let κ be a cardinal, and write [X] n for the set of subsets of X of size n. Given an ordinal β and ordinals αi for all i ∈ κ, write β →top (αi) n i∈κ to mean that for every function c : [β] n → κ (a colouring) there is some subspace X ⊆ β and some i ∈ κ such that X is homeomorphic to αi and [X] n ⊆ c −1 ({i}). We examine the cases n = 1 and n = 2, defining the topological pigeonhole number P top (αi) i∈κ to be the least ordinal β (when one exists) such that β →top (αi) 1 i∈κ , and the topological Ramsey number Rtop (αi) i∈κ to be the least ordinal β (when one exists) such that β →top (αi) 2 i∈κ . We resolve the case n = 1 by determining the topological pigeonhole number of an arbitrary sequence of ordinals, including an independence result for one class of cases. In the case n = 2, we prove a topological version of the Erd˝os–Milner theorem, namely that Rtop (α, k) is countable whenever α is countable and k is finite. More precisely, we prove that Rtop(ω ω β , k + 1) ≤ ω ω β·k for all countable ordinals β and all positive integers k. We also provide more careful upper bounds for certain small ordinals, including Rtop(ω + 1, k + 1) = ω k + 1, Rtop(α, k) < ωω whenever α < ω2 , Rtop(ω 2 , k) ≤ ω ω and Rtop(ω 2 + 1, k + 2) ≤ ω ω·k + 1 for all positive integers k. Outside the partition calculus, we prove a topological analogue of Hausdorff’s theorem on scattered total orderings. This allows us to characterise countable subspaces of ordinals as the order topologies of countable scattered total orderings. As an application, we compute the number of subspaces of an ordinal up to homeomorphism. Finally, we study the group of autohomeomorphisms of ω n ·m+1 for finite n and m. We classify the normal subgroups contained in the pointwise stabiliser of the limit points. These subgroups fall naturally into D (n) disjoint sets, each either countable or of size 22 ℵ0 , where D (n) is the number of ⊆-antichains of P ({1, 2, . . . , n}). Our techniques span a variety of disciplines, including set theory, general topology and permutation group theor
Higher Segal spaces I
This is the first paper in a series on new higher categorical structures
called higher Segal spaces. For every d > 0, we introduce the notion of a
d-Segal space which is a simplicial space satisfying locality conditions
related to triangulations of cyclic polytopes of dimension d. In the case d=1,
we recover Rezk's theory of Segal spaces. The present paper focuses on 2-Segal
spaces. The starting point of the theory is the observation that Hall algebras,
as previously studied, are only the shadow of a much richer structure governed
by a system of higher coherences captured in the datum of a 2-Segal space. This
2-Segal space is given by Waldhausen's S-construction, a simplicial space
familiar in algebraic K-theory. Other examples of 2-Segal spaces arise
naturally in classical topics such as Hecke algebras, cyclic bar constructions,
configuration spaces of flags, solutions of the pentagon equation, and mapping
class groups.Comment: 221 page
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