M.Sc Thesis | |

M.Sc Student | Algom Kfir Yael |
---|---|

Subject | Diagram Groups and Their Action on Farley Complexes |

Department | Department of Mathematics |

Supervisors | PROFESSOR EMERITUS Bronislaw Wajnryb |

PROF. Michah Sageev |

Intuitively, spherical
diagrams over a semigroup presentation
*
P*
are
maps *D: S ^{2} → *

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 *K _{g}*
which can be but is not always finite dimensional. The infimum of