4 research outputs found
Nonexistence of embeddings with uniformly bounded distortions of Laakso graphs into diamond graphs
Diamond graphs and Laakso graphs are important examples in the theory of
metric embeddings. Many results for these families of graphs are similar to
each other. In this connection, it is natural to ask whether one of these
families admits uniformly bilipschitz embeddings into the other. The well-known
fact that Laakso graphs are uniformly doubling but diamond graphs are not,
immediately implies that diamond graphs do not admit uniformly bilipschitz
embeddings into Laakso graphs. The main goal of this paper is to prove that
Laakso graphs do not admit uniformly bilipschitz embeddings into diamond
graphs
A characterization of superreflexivity through embeddings of lamplighter groups
We prove that finite lamplighter groups
with a standard set of generators
embed with uniformly bounded distortions into any non-superreflexive Banach
space, and therefore form a set of test-spaces for superreflexivity. Our proof
is inspired by the well known identification of Cayley graphs of infinite
lamplighter groups with the horocyclic product of trees. We cover
by three sets with a structure similar to a
horocyclic product of trees, which enables us to construct well-controlled
embeddings
Lipschitz free spaces on finite metric spaces
Main results of the paper:
(1) For any finite metric space the Lipschitz free space on contains
a large well-complemented subspace which is close to .
(2) Lipschitz free spaces on large classes of recursively defined sequences
of graphs are not uniformly isomorphic to of the corresponding
dimensions. These classes contain well-known families of diamond graphs and
Laakso graphs.
Interesting features of our approach are: (a) We consider averages over
groups of cycle-preserving bijections of graphs which are not necessarily graph
automorphisms; (b) In the case of such recursive families of graphs as Laakso
graphs we use the well-known approach of Gr\"unbaum (1960) and Rudin (1962) for
estimating projection constants in the case where invariant projections are not
unique
Metric dimension reduction: A snapshot of the Ribe program
The purpose of this article is to survey some of the context, achievements,
challenges and mysteries of the field of metric dimension reduction, including
new perspectives on major older results as well as recent advances.Comment: proceedings of ICM 201