Timothy Riley

Higher connectedness of asymptotic cones

We give coarse geometric conditions for a metric space X to have N-connected asymptotic cones. These conditions are expressed in terms of certain filling functions concerning filling N-spheres in an appropriate coarse sense. We interpret the criteria in the case where X is a finitely generated group with a word metric. This leads to upper bounds on filling functions for groups with simply connected cones – in particular they have linearly bounded filling length functions. We prove that if all the asymptotic cones of are N-connected then is of type FN+1 and we provide N-th order isoperimetric and isodiametric functions. Also we show that the asymptotic cones of a virtually polycyclic group are all co ntractible if and only if is virtually nilpotent.

A finitely presented group with unbounded dead-end depth

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 elements.

THE UNBOUNDED DEAD-END DEPTH PROPERTY IS NOT A GROUP INVARIANT

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

Palindromic width of wreath products, metabelian groups, and max-n solvable groups

Abstract 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

The Dehn Function of Stallings’ Group

G \wr \mathbb {Z}^{r}

Some duality conjectures for finite graphs and their group theoretic consequences

. We also give a new, self-contained proof that finitely generated metabelian groups have finite palindromic width. Finally, we show that solvable groups satisfying the maximal condition on normal subgroups (max-n) have finite palindromic width.

Erratum to “A finitely presented group with unbounded dead-end depth”

We prove that the Dehn function of a group of Stallings that is finitely presented but not of type

Novel technique for isolated accessory right heart transplantation for congenital heart disease

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