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The dead-end depth of an element g of a group with finite generating set A is the distance from g to the complement of the radius d A (1, g) closed ball, in the word metric d A. We exhibit a finitely presented group K with two finite generating sets A and B such that dead-end depth is unbounded on K with respect to A but is bounded above by two with respect… (More)

A group has finite palindromic width if there exists n such that every element can be expressed as a product of n or fewer palindromic words. We show that if G has finite palindromic width with respect to some generating set, then so does G Z r. We also give a new, self-contained, proof that finitely generated metabelian groups have finite palindromic… (More)

We show that for integers k ≥ 2 and n ≥ 3, the diameter of the Cayley graph of SLn(Z/kZ) associated to a standard two-element generating set, is at most a constant times n 2 ln k. This answers a question of A. Lubotzky concerning SLn(Fp) and is unexpected because these Cayley graphs do not form an expander family. Our proof amounts to a quick algorithm for… (More)

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