Finite simple groups of Lie type as expanders

Alexander Lubotzky

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Gruenhage asked if it was possible to cover the real line by less than continuum many translates of a compact nullset. Under the Continuum Hypothesis the answer is obviously negative. Elekes and Stepr mans gave an affirmative answer by showing that if $C_{E K}$ is the well known compact nullset considered first by Erdős and Kakutani then ℝ can be covered by cof() many translates of $C_{E K}$ . As this set has no analogue in more general groups, it was asked by Elekes and Stepr mans whether such a result...

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ContentsIntroduction.................................................................................................................... 31. Preliminaries (topology measure).................................................................... 32. Problems and the theorem.................................................................................... 73. Preliminaries (abstract groups, Cartesian products)....................................... 94. Preliminaries (automorphisms, duality...

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In this Note, we define a class of stratified Lie groups of arbitrary step (that are called “groups of type $☆$ ” throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type $☆$ is rectifiable in the intrinsic sense (De Giorgi’s rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano...

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It is proved that if $A$ is an abelian $p$ -group with a pure subgroup $G$ so that $A / G$ is at most countable and $G$ is either $p^{ω + n}$ -totally projective or $p^{ω + n}$ -summable, then $A$ is either $p^{ω + n}$ -totally projective or $p^{ω + n}$ -summable as well. Moreover, if in addition $G$ is nice in $A$ , then $G$ being either strongly $p^{ω + n}$ -totally projective or strongly $p^{ω + n}$ -summable implies that so is $A$ . This generalizes a classical result of Wallace (J. Algebra, 1971) for totally projective $p$ -groups as well as continues our recent investigations...

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