| A quasi-isometry is a map between metric spaces which distorts distance by a bounded amount. We view a finitely generated group as a metric space via the word metric on the associated Cayley graph. In 1983 Gromov proposed a program of studying and classifying all finitely generated groups up to quasi-isometry. The two main types of results in this program are for lattices in semisimple Lie groups and for the solvable Baumslag-Solitar groups, which are not lattices in any Lie group. The results stated below are for the projective special linear groups ;The first two theorems together give the first example of a class of groups which have the same quasi-isometry group but are not quasi-isometric. The quasi-isometry group of a space X is the set of all self quasi-isometries of X, modulo those a bounded distance from the identity in the sup norm, under composition of quasi-isometries. (1)... |
