On a Generalization of Dehn's Algorithm

Abstract

Viewing Dehn's algorithm as a rewriting system, we generalize to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include finitely generated nilpotent groups, many relatively hyperbolic groups including geometrically finite groups and fundamental groups of certain geometrically decomposable 3-manifolds. The class has several nice closure properties. We also show that if a group has an infinite subgroup and one of exponential growth, and they commute, then it does not admit such an algorithm. We dub these Cannon's algorithms.

Groups That Do and Do Not Have Growing Context-Sensitive Word Problem

“…We generalize results of [6] to find many examples of groups not admitting non-deterministic Cannon's algorithms. This adds to the examples of Kambites and Otto in [7] of groups separating context-sensitive and growing context-sensitive word problems, and provides a new language-theoretic separation result.…”

“…The Cannon's algorithms described in [6] are deterministic, but there was some brief discussion at the end of [6, Sec. 1.3] of non-deterministic generalizations, and the close connections between groups with non-deterministic Cannon's algorithms and those with growing context-sensitive word problem.…”

“…2 below.) It was clear that an appropriate modification of the theorems in [6, Sec. 7] should imply that various groups, including direct products F m × F n of free groups with m > 1 and n ≥ 1 and other examples mentioned below, had word problems that were context-sensitive but not growing context-sensitive.…”

“…Two different versions of Cannon's algorithms, known as incremental and non-incremental , are defined in [6], and both of these are deterministic. In [6, Proposition 4] it is shown that the language of an incremental Cannon's algorithm is also the language of a non-incremental Cannon's algorithm.…”

“… We prove that a group has word problem that is a growing context-sensitive language precisely if its word problem can be solved using a non-deterministic Cannon's algorithm (the deterministic algorithms being defined by Goodman and Shapiro in [6]). We generalize results of [6] to find many examples of groups not admitting non-deterministic Cannon's algorithms.…”