A graph X is defined inductively to be (a0, . . . , an−1)-regular if X is a0-regular and for every vertex v of X, the sphere of radius 1 around v is an (a1, . . . , an−1)-regular graph. Such a graph X is said to be highly regular (HR) of level n if an−1 = 0. Chapman, Linial and Peled [CLP20] studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders "globally and locally". They ask whether such HR-graphs of level 3 exist.In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can lead to such graphs. Given a Coxeter system (W, S) and a subset M of S, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope PW,M , which form an infinite family of expander graphs when (W, S) is indefinite and PW,M has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of (W, S). The expansion stems from applying superapproximation to the congruence subgroups of the linear group W .This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked in [CLP20].
A. We generalize a result of Sury [Sur92] and prove that uniform discreteness of cocompact lattices in higher rank semisimple Lie groups (first conjectured by Margulis [Mar91]) is equivalent to a weak form of Lehmer's conjecture. We include a short survey of related results and conjectures.
In this paper we prove explicit estimates for the size of small lifts of points in homogeneous spaces. Our estimates are polynomially effective in the volume of the space and the injectivity radius.
The objective of this paper is to determine the lattices of minimal covolume in prefixSLnfalse(double-struckRfalse), for n⩾3. The answer turns out to be the simplest one: prefixSLnfalse(double-struckZfalse) is, up to automorphism, the unique lattice of minimal covolume in prefixSLnfalse(double-struckRfalse). In particular, lattices of minimal covolume in prefixSLnfalse(double-struckRfalse) are non‐uniform when n⩾3, contrasting with Siegel's result for prefixSL2false(double-struckRfalse). This answers for prefixSLnfalse(double-struckRfalse) the question of Lubotzky: is a lattice of minimal covolume typically uniform or not?
A graph X is defined inductively to be (a0, . . . , an−1)-regular if X is a0-regular and for every vertex v of X, the sphere of radius 1 around v is an (a1, . . . , an−1)-regular graph. Such a graph X is said to be highly regular (HR) of level n if an−1 = 0. Chapman, Linial and Peled [CLP20] studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders "globally and locally", and asked about the existence of HR-graphs of level 3.In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can be used to construct such graphs. Given a Coxeter system (W, S) and a subset M of S, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope PW,M , which form an infinite family of expander graphs when (W, S) is indefinite and PW,M has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of (W, S). The expansion stems from applying superapproximation to the congruence subgroups of the linear group W .This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked in [CLP20].
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