7,151 research outputs found
On the Classification of Vertex-Transitive Structures
We consider the classification problem for several classes of countable structures which are “vertex-transitive”, meaning that the automorphism group acts transitively on the elements. (This is sometimes called homogeneous.) We show that the classification of countable vertex-transitive digraphs and partial orders are Borel complete. We identify the complexity of the classification of countable vertex-transitive linear orders. Finally we show that the classification of vertex-transitive countable tournaments is properly above E0 in complexity
Classification of Vertex-Transitive Structures
When one thinks of objects with a significant level of symmetry it is natural to expect there to be a simple classification. However, this leads to an interesting problem in that research has revealed the existence of highly symmetric objects which are very complex when considered within the framework of Borel complexity. The tension between these two seemingly contradictory notions leads to a wealth of natural questions which have yet to be answered.
Borel complexity theory is an area of logic where the relative complexities of classification problems are studied. Within this theory, we regard a classification problem as an equivalence relation on a Polish space. An example of such is the isomorphism relation on the class of countable groups. The notion of a Borel reduction allows one to compare complexities of various classification problems.
The central aim of this research is determine the Borel complexities of various classes of vertex-transitive structures, or structures for which every pair or elements are equivalent under some element of its automorphism group. John Clemens has shown that the class of vertex-transitive graphs has maximum possible complexity, namely Borel completeness. On the other hand, we show that the class of vertex-transitive linear orderings does not.
We explore this phenomenon further by considering other natural classes of vertex-transitive structures such as tournaments and partial orderings. In doing so, we discover that several other complexities arise for classes of vertex-transitive structures
Polygonal Complexes and Graphs for Crystallographic Groups
The paper surveys highlights of the ongoing program to classify discrete
polyhedral structures in Euclidean 3-space by distinguished transitivity
properties of their symmetry groups, focussing in particular on various aspects
of the classification of regular polygonal complexes, chiral polyhedra, and
more generally, two-orbit polyhedra.Comment: 21 pages; In: Symmetry and Rigidity, (eds. R.Connelly, A.Ivic Weiss
and W.Whiteley), Fields Institute Communications, to appea
Polyhedra, Complexes, Nets and Symmetry
Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are
finite or infinite 3-periodic structures with interesting geometric,
combinatorial, and algebraic properties. They can be viewed as finite or
infinite 3-periodic graphs (nets) equipped with additional structure imposed by
the faces, allowed to be skew, zig-zag, or helical. A polyhedron or complex is
"regular" if its geometric symmetry group is transitive on the flags (incident
vertex-edge-face triples). There are 48 regular polyhedra (18 finite polyhedra
and 30 infinite apeirohedra), as well as 25 regular polygonal complexes, all
infinite, which are not polyhedra. Their edge graphs are nets well-known to
crystallographers, and we identify them explicitly. There also are 6 infinite
families of "chiral" apeirohedra, which have two orbits on the flags such that
adjacent flags lie in different orbits.Comment: Acta Crystallographica Section A (to appear
Classification of some countable descendant-homogeneous digraphs
For finite q, we classify the countable, descendant-homogeneous digraphs in
which the descendant set of any vertex is a q-valent tree. We also give
conditions on a rooted digraph G which allow us to construct a countable
descendant-homogeneous digraph in which the descendant set of any vertex is
isomorphic to G.Comment: 16 page
Countable locally 2-arc-transitive bipartite graphs
We present an order-theoretic approach to the study of countably infinite
locally 2-arc-transitive bipartite graphs. Our approach is motivated by
techniques developed by Warren and others during the study of cycle-free
partial orders. We give several new families of previously unknown countably
infinite locally-2-arc-transitive graphs, each family containing continuum many
members. These examples are obtained by gluing together copies of incidence
graphs of semilinear spaces, satisfying a certain symmetry property, in a
tree-like way. In one case we show how the classification problem for that
family relates to the problem of determining a certain family of highly
arc-transitive digraphs. Numerous illustrative examples are given.Comment: 29 page
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