UCL London

I will describe a method to find negatively curved structures on some groups, by manipulating metrics on piecewise hyperbolic complexes. As an example, I will prove that hyperbolic limit groups are CAT(-1).

We formulate a new theory for equilibria of 2-braids, i.e., structures

formed by two elastic rods winding around each other in continuous contact

and subject to a local interstrand interaction. Unlike in previous work no

assumption is made on the shape of the contact curve. The theory is developed

in terms of a moving frame of directors attached to one of the strands with

one of the directors pointing to the position of the other strand. The

constant-distance constraint is automatically satisfied by the introduction

of what we call braid strains. The price we pay is that the potential energy

involves arclength derivatives of these strains, thus giving rise to a

second-order variational problem. The Euler-Lagrange equations for this

problem (in Euler-Poincare form) give balance equations for the overall

braid force and moment referred to the moving frame as well as differential

equations that can be interpreted as effective constitutive relations

encoding the effect that the second strand has on the first as the braid

deforms under the action of end loads. Hard contact models are used to obtain

the normal contact pressure between strands that has to be non-negative for

a physically realisable solution without the need for external devices such

as clamps or glue to keep the strands together. The theory is first

illustrated by a few simple examples and then applied to several problems

that require the numerical solution of boundary-value problems. Both open

braids and closed braids (links and knots) are considered and current

applications to DNA supercoiling are discussed.