Roger Lyndon, born on Decl8, 19I7 in Calais (Maine, USA), entered Harvard University in 1935 with the aim of studying literature and becoming a writer. However, when he discovered that, for him, mathematics required less effort than literature, he switched and graduated from Harvard in 1939. After completing his Master's Degree in 1941, he taught at Georgia Tec h, th en returned to Harvard in 1942 and there taugh1 navigation to pilots while, supervised by S. MacLane, he studied for his Ph.D., awarded in 1946 for a thesis entitled The Cohomology Theory of Group Extensions. Influenced by Tarski, Lyndon was later to work on model th eo ry. Accepti ng a position at Princeton, Ralph Fox and Reideme ister's visit in 1948 were major influencea on him to work in combinatorial group theory. In 1953 Lyndon left Princeton for a chair at the Universit y of Michigan where he then remained except fo r visiting professorships at Berkeley, London, Montpellier and Amiens. Lyndon made numerous major contributions to combinatorial group theory. These included the development of "small cancellation theory", his introduction ofuaspherical" presentations of groups and his work on length functions. He died on June 8, 1988.Paul Schupp, born on March 12, 1937 in Cleveland, Ohio was a student of Roger Lyndon'S at the University of Michigan where he wrote a thesis ofuDehn's Algorithm and the Conjugacy Problem". After a year at the University of Wisconsin he moved to the University of Illinois where he remained. For several years he was also concurrently Visi ting Professor at the Un iversity Paris VII and a member of the Labora toire d'informatique TMorique et Programmation (fou nded by M. P. Schutzenberger). Schupp further developed the use of cancellation diagrams in com· bill.atorial group theory, introducing conjugacy diagrams, diagrams on com pact surfaces, diagrams over free products with amalgam· ation and HNN extensions and applications to Arlin groups. He then worked with David Mulle r on connections between group theory and formal language theory and on the theory of finite automata on infinite inputs. His current interest is using geometric methods to investigate the computational complexity of algorithms in combinatorial group theory.

No abstract

We give a precise definition of "generic-case complexity" and show that for a very large class of finitely generated groups the classical decision problems of group theory -the word, conjugacy and membership problems -all have linear-time generic-case complexity. We prove such theorems by using the theory of random walks on regular graphs.

No abstract

We prove that Whitehead's algorithm for solving the automorphism problem in a fixed free group F k has strongly linear time generic-case complexity. This is done by showing that the "hard" part of the algorithm terminates in linear time on an exponentially generic set of input pairs. We then apply these results to one-relator groups. We obtain a Mostow-type isomorphism rigidity result for random one-relator groups: If two such groups are isomorphic then their Cayley graphs on the given generating sets are isometric. Although no nontrivial examples were previously known, we prove that one-relator groups are generically complete groups, that is, they have trivial center and trivial outer automorphism group. We also prove that the stabilizers of generic elements of F k in Aut(F k ) are cyclic groups generated by inner automorphisms and that Aut(F k )-orbits are uniformly small in the sense of their growth entropy. We further prove that the number I k (n) of isomorphism types of k-generator one-relator groups with defining relators of length n satisfies c 1 n (2k − 1)where c 1 , c 2 are positive constants depending on k but not on n. Thus I k (n) grows in essentially the same manner as the number of cyclic words of length n.