160 research outputs found
Classification of flag-transitive Steiner quadruple systems
A Steiner quadruple system of order v is a 3-(v,4,1) design, and will be
denoted SQS(v). Using the classification of finite 2-transitive permutation
groups all SQS(v) with a flag-transitive automorphism group are completely
classified, thus solving the "still open and longstanding problem of
classifying all flag-transitive 3-(v,k,1) designs" for the smallest value of k.
Moreover, a generalization of a result of H. Lueneburg (1965, Math. Z. 89,
82-90) is achieved.Comment: 11 page
A Census Of Highly Symmetric Combinatorial Designs
As a consequence of the classification of the finite simple groups, it has
been possible in recent years to characterize Steiner t-designs, that is
t-(v,k,1) designs, mainly for t = 2, admitting groups of automorphisms with
sufficiently strong symmetry properties. However, despite the finite simple
group classification, for Steiner t-designs with t > 2 most of these
characterizations have remained longstanding challenging problems. Especially,
the determination of all flag-transitive Steiner t-designs with 2 < t < 7 is of
particular interest and has been open for about 40 years (cf. [11, p. 147] and
[12, p. 273], but presumably dating back to 1965). The present paper continues
the author's work [20, 21, 22] of classifying all flag-transitive Steiner
3-designs and 4-designs. We give a complete classification of all
flag-transitive Steiner 5-designs and prove furthermore that there are no
non-trivial flag-transitive Steiner 6-designs. Both results rely on the
classification of the finite 3-homogeneous permutation groups. Moreover, we
survey some of the most general results on highly symmetric Steiner t-designs.Comment: 26 pages; to appear in: "Journal of Algebraic Combinatorics
Singer quadrangles
[no abstract available
Quasi-Isometric Embeddings of Symmetric Spaces
We prove a rigidity theorem that shows that, under many circumstances,
quasi-isometric embeddings of equal rank, higher rank symmetric spaces are
close to isometric embeddings. We also produce some surprising examples of
quasi-isometric embeddings of higher rank symmetric spaces. In particular, we
produce embeddings of into when no
isometric embeddings exist. A key ingredient in our proofs of rigidity results
is a direct generalization of the Mostow-Morse Lemma in higher rank. Typically
this lemma is replaced by the quasi-flat theorem which says that maximal
quasi-flat is within bounded distance of a finite union of flats. We improve
this by showing that the quasi-flat is in fact flat off of a subset of
codimension .Comment: Exposition improved, outlines of proofs added to introduction. Typos
corrected, references added. Also some discussion of the reducible case adde
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