Aner Shalev

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We prove that random d-regular Cayley graphs of the symmetric group asymp-totically almost surely have girth at least (log d−1 |G|) 1/2 /2 and that random d-regular Cay-ley graphs of simple algebraic groups over F q asymptotically almost surely have girth at least log d−1 |G|/ dim(G). For the symmetric p-groups the girth is between log log |G| and (log |G|)(More)
We construct a sequence of groups G n , and explicit sets of generators Y n ⊂ G n , such that all generating sets have bounded size, and the associated Cayley graphs are all expanders. The group G 1 is the alternating group A d , the set of even permutations on the elements {1, 2,. .. , d}. The group G n is the group of all even symmetries of the rooted(More)
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