Volume 4, Issue 2

Abstract. This paper is a survey of known results (old and new) about equations over groups, mainly solvable groups. Current directions of research are discussed in some detail. A number of open questions are included.

Abstract. Results on algorithmic problems on strings that are given in a compressed form via straight-line programs are surveyed. A straight-line program is a context-free grammar that generates exactly one string. In this way, exponential compression rates can be achieved. Among others, we study pattern matching for compressed strings, membership problems for compressed strings in various kinds of formal languages, and the problem of querying compressed strings. Applications in combinatorial group theory and computational topology and to the solution of word equations are discussed as well. Finally, extensions to compressed trees and pictures are considered.

Abstract. We investigate the cogrowth and distribution of geodesics in R. Thompson's group F .

Abstract. Cyclic words are equivalence classes of cyclic permutations of ordinary words. When a group is given by a rewriting relation, a rewriting system on cyclic words is induced, which is used to construct algorithms to find minimal length elements of conjugacy classes in the group. These techniques are applied to the universal groups of Stallings pregroups and in particular to free products with amalgamation, HNN-extensions and virtually free groups, to yield simple and intuitive algorithms and proofs of conjugacy criteria.

Abstract. A cylinder is the set of infinite words with fixed prefix u . A double-cylinder is “the same” for bi-infinite words. We show that for every word u and any automorphism of the free group F the image is a finite union of cylinders. The analogous statement is true for double cylinders. We give (a) an algorithm, and (b) a precise formula which allows one to determine this finite union of cylinders.

Abstract. The objective of this work is to survey several digital signatures proposed in the last decade using non-commutative groups and rings and propose a digital signature using non-commutative groups and analyze its security.