Recent work of Gromov, Epstein, Cannon, Thurston and many others has generated strong interest in the geometric and algorithmic structure of finitely generated infinite groups. (See [16],[17] and [14].) Many of these structures are preserved by taking graph products. The graph product of groups (not to be confused with the fundamental group of a graph of groups) is a product mixing direct and free products. Whether the product between two groups in the graph product is free or direct is determined by a simplicial graph. Given a simplicial graph we say that two vertices are adjacent if they are joined by a single edge. Definition. Given a finite simplicial graph G with a group (or monoid) attached to each vertex, the associated graph product is the group (monoid) generated by each of the vertex groups (monoids) with the added relations that elements of distinct adjacent vertex groups commute. Graph products were defined by Green [15], and have also been studied by Chiswell [8], [9], [10]. Graph products are a generalization of "semifree groups"

Autostackability for finitely generated groups is defined via a topological property of the associated Cayley graph which can be encoded in a finite state automaton. Autostackable groups have solvable word problem and an effective inductive procedure for constructing van Kampen diagrams with respect to a canonical finite presentation. A comparison with automatic groups is given. Another characterization of autostackability is given in terms of prefix-rewriting systems. Every group which admits a finite complete rewriting system or an asynchronously automatic structure with respect to a prefix-closed set of normal forms is also autostackable. As a consequence, the fundamental group of every closed 3-manifold with any of the eight possible uniform geometries is autostackable.

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We study the regularity of several languages derived from conjugacy classes in a finitely generated group G for a variety of examples including word hyperbolic, virtually abelian, Artin, and Garside groups. We also determine the rationality of the growth series of the shortlex conjugacy language in virtually cyclic groups, proving one direction of a conjecture of Rivin.

We prove that a finitely generated group G is virtually free if and only if there exists a generating set for G and k > 0 such that all k-locally geodesic words with respect to that generating set are geodesic. Mathematics Subject Classification (2000). 20E06; Secondary: 20F67.