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- M. R. Bridsonand, T. R. Riley
- 2005

The diameter of a disc filling a loop in the universal covering of a Rie-mannian manifold M may be measured extrinsically using the distance function on M or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram ∆ filling a word that represents 1 in a finitely presented group Γ can either be… (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)

- Sean Clearyand, Tim R. Riley
- 2004

The dead-end depth of an element g of a group G, with respect to a 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 defined with respect to A. We exhibit a finitely presented group G with a finite generating set with respect to which there is no upper bound on the dead-end depth of… (More)

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)

- M R Bridson, T R Riley
- 2005

A. The filling length of an edge-circuit η in the Cayley 2-complex of a finitely presented group is the least integer L such that there is a combinatorial null-homotopy of η down to a basepoint through loops of length at most L. We introduce similar notions in which the null-homotopy is not required to fix a basepoint, and in which the contracting… (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)

- Centre De Recerca, Mathemàtica Barcelona, T R Riley
- 2006

- Tim R. Riley
- 2004

The dead end depth of an element g of a group G 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 defined with respect to A. We say that G has infinite dead end depth when dead end depth is unbounded, ranging over G. We exhibit a finitely presented group G with a finite… (More)

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