7,010 research outputs found
Expanders, rank and graphs of groups
Let G be a finitely presented group, and let {G_i} be a collection of finite
index normal subgroups that is closed under intersections. Then, we prove that
at least one of the following must hold: 1. G_i is an amalgamated free product
or HNN extension, for infinitely many i; 2. the Cayley graphs of G/G_i (with
respect to a fixed finite set of generators for G) form an expanding family; 3.
inf_i (d(G_i)-1)/[G:G_i] = 0, where d(G_i) is the rank of G_i.
The proof involves an analysis of the geometry and topology of finite Cayley
graphs. Several applications of this result are given.Comment: 13 pages; to appear in Israel J. Mat
Expanders and right-angled Artin groups
The purpose of this article is to give a characterization of families of
expander graphs via right-angled Artin groups. We prove that a sequence of
simplicial graphs forms a family of expander
graphs if and only if a certain natural mini-max invariant arising from the cup
product in the cohomology rings of the groups
agrees with the Cheeger constant of the
sequence of graphs, thus allowing us to characterize expander graphs via
cohomology. This result is proved in the more general framework of \emph{vector
space expanders}, a novel structure consisting of sequences of vector spaces
equipped with vector-space-valued bilinear pairings which satisfy a certain
mini-max condition. These objects can be considered to be analogues of expander
graphs in the realm of linear algebra, with a dictionary being given by the cup
product in cohomology, and in this context represent a different approach to
expanders that those developed by Lubotzky-Zelmanov and Bourgain-Yehudayoff.Comment: 21 pages. Accepted version. To appear in J. Topol. Ana
Suzuki groups as expanders
We show that pairs of generators for the family Sz(q) of Suzuki groups may be
selected so that the corresponding Cayley graphs are expanders. By combining
this with several deep works of Kassabov, Lubotzky and Nikolov, this
establishes that the family of all non-abelian finite simple groups can be made
into expanders in a uniform fashion.Comment: 17 page
Trivalent expanders and hyperbolic surfaces
We introduce a family of trivalent expanders which tessellate compact
hyperbolic surfaces with large isometry groups. We compare this family with
Platonic graphs and modifications of them and prove topological and spectral
properties of these families
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