M.Sc Student Algom Kfir Yael Diagram Groups and Their Action on Farley Complexes Department of Mathematics Professor Emeritus Bronislaw Wajnryb Professor Michah Sageev

Abstract

Intuitively, spherical diagrams over a semigroup presentation P are maps D: S2 C(P ) (where C(P)  is the Cayley graph of the presentation), modulo some equivalence relation. A diagram group D( P ,w) over a semigroup presentation P is comprised of all spherical diagrams whose image contains a fixed base path w with the operation of concatenation along this path.

The class of diagram groups contains free groups of arbitrary rank, free abelian groups of finite rank, and the well known Thompson's group F . It is closed under direct products and arbitrary free products. V. Guba and M. Sapir in [GS96] prove many nice properties of diagram groups such as: there is a fast algorithm for solving the word problem for any diagram group, if P has a solvable word problem then has a solvable conjugacy problem, if P is finite then D(P,w) is finitely presented.

In [Far04] D. Farley proved that any diagram group D(P,w), over a finite semigroup presentation P acts properly and freely by isometries on a CAT(0) proper cube complex K( P,w). He conjectured that every element g in D(P,w) is a hyperbolic isometry. Explicitly, τ(g) = inf{ d(x,gx)  | x in K(P,w) } is positive and attained.

For every element g in D(P,w) we have found a convex g-invarient subspace Kg  which can be but is not always finite dimensional. The infimum of d(x,gx) on this subspace is the same as the infimum on the whole space K(P,w). If  Kg is finite dimensional then this result implies that g is hyperbolic. Futheremore, for every g we have found a finite set of vertices Jg which are moved most efficiently compared with all the other vertices thus inf{ d(x,gx)  | x  is a vertex in K(P,w) } is attained.