The proposed research is in Geometric Group Theory, specifically on problems pertaining to the geometry, topology and asymptotics of the word problem, particularly along avenues initiated by Gromov in his seminal book "Asymptotic Invariants of Infinite Groups." A first direction concerns isoperimetric and filling length functions, a meeting point of geometry and algorithmic complexity. Examples of problems for attack here are whether Cohen's Double Exponential Theorem is sharp, and to establish long standing claims of Thurston about isoperimetric functions of the Special Linear Groups. Also addressed in the proposal is a question of Gromov about the uniqueness of a limit of a group viewed from increasingly distant vantage points (an asymptotic cone); a line of attack is being pursued in collaboration with M.R. Bridson. Another area is the "grammar of the Word Problem", where geometry meets linguistic complexity. The P.I. proposes to investigate the combings of nilpotent groups as well as applications to the Andrews-Curtis Conjecture. The proposal includes collaborative work and it is anticipated that the inter-disciplinary nature of the work will enable additional collaborative efforts both in the United States and abroad.
The Word Problem was posed by Max Dehn in 1912, who presciently described it as a "fundamental problem whose solution is very difficult and which will not be possible without a penetrating study of the subject." The Word Problem for a group (a fundamental algebraic object capturing symmetries) asks for a systematic method (in modern terms, an algorithm) which, given a finite list (a "word") of basic symmetries (distinguished group elements known as "generators"), declares whether or not their product is the identity (the trivial symmetry that moves nothing). One of the crowning achievements of 20th century mathematics was the construction by Boone and Novikov of groups for which no such algorithm can exist. However, that is by no means the end of the story. The Word Problem transcends its origins in algebra, decidability and complexity theory, and turns out to be a bridge to geometry and topology, as well as to linguistic complexity. To a group we can associate a topological space and vice versa. Recognising a word to be trivial amounts to spanning a loop with a disc in a corresponding space, and geometrical features of this disc (such as area) translate to algorithmic complexity measures (such as time) of the Word Problem. Moreover, words over a group form grammars whose linguistic features relate to the ease of navigating around the corresponding space. Thus there are tantalising links between minimal surfaces (soap films) and algorithmic complexity, and between geometry of spaces and grammars. These are the connections the P.I. proposes to develop and exploit.
