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By work of Belyi, the absolute Galois group G_Q of the rational numbers embeds into a subgroup \hat{GT} called the Grothendeick-Teichmuller group of the group A of continuous automorphisms of a profinite group on two generators. We show that a rich class of representations of G_Q lifts to \hat{GT} by showing they lift all the way to a finite index subgroup of A.more » « less
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Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. In recent years, a high-dimensional theory has emerged. In this paper, these developments are surveyed. After explaining their connection to the Ramanujan conjecture, we will present some old and new results with an emphasis on random walks on these discrete objects and on the Euclidean spheres. The latter lead to ‘golden gates’ which are of importance in quantum computation. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.more » « less
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