10,504 research outputs found
Consequences of some outerplanarity extensions
In this expository paper we revise some extensions of Kuratowski planarity criterion, providing a link between the embeddings of infinite graphs without accumulation points and the embeddings of finite graphs with some
distinguished vertices in only one face. This link is valid for any surface and for some pseudosurfaces. On the one hand, we present some key ideas that are not easily accessible. On the other hand, we state the relevance
of infinite, locally finite graphs in practice and suggest some ideas for future research
Metric characterizations of superreflexivity in terms of word hyperbolic groups and finite graphs
We show that superreflexivity can be characterized in terms of bilipschitz
embeddability of word hyperbolic groups. We compare characterizations of
superreflexivity in terms of diamond graphs and binary trees. We show that
there exist sequences of series-parallel graphs of increasing topological
complexity which admit uniformly bilipschitz embeddings into a Hilbert space,
and thus do not characterize superreflexivity
All reducts of the random graph are model-complete
We study locally closed transformation monoids which contain the automorphism
group of the random graph. We show that such a transformation monoid is locally
generated by the permutations in the monoid, or contains a constant operation,
or contains an operation that maps the random graph injectively to an induced
subgraph which is a clique or an independent set. As a corollary, our
techniques yield a new proof of Simon Thomas' classification of the five closed
supergroups of the automorphism group of the random graph; our proof uses
different Ramsey-theoretic tools than the one given by Thomas, and is perhaps
more straightforward. Since the monoids under consideration are endomorphism
monoids of relational structures definable in the random graph, we are able to
draw several model-theoretic corollaries: One consequence of our result is that
all structures with a first-order definition in the random graph are
model-complete. Moreover, we obtain a classification of these structures up to
existential interdefinability.Comment: Technical report not intended for publication in a journal. Subsumed
by the more recent article 1003.4030. Length 14 pages
Scales for co-compact embeddings of virtually free groups
Let be a group which is virtually free of rank at least 2 and let
be the family of totally disconnected, locally
compact groups containing as a co-compact lattice.
We prove that the values of the scale function with respect to groups in
evaluated on the subset have only finitely
many prime divisors. This can be thought of as a uniform property of the family
.Comment: 12 pages; key words: uniform lattice, virtually free group, totally
disconnected group, scale function (Error in references corrected in version
2
A proof of the rooted tree alternative conjecture
Bonato and Tardif conjectured that the number of isomorphism classes of trees
mutually embeddable with a given tree T is either 1 or infinite. We prove the
analogue of their conjecture for rooted trees. We also discuss the original
conjecture for locally finite trees and state some new conjectures
- …