context word problem does not depend on the choice of a finite marked generating set for the group, and such a group is finitely presentable. Muller and Schupp then show, using the context-free grammar for the language , that the Cayley graph of G with respect to X is K-triangulable for some integer K>0. This means that every closed path in can be, by adding several ``diagonals", decomposed into triangles in such a way that the label of every triangle is a relation in G of length at most K over X. They then use this triangulability property of the Cayley graph to show that either G is a finite group, or G has more than one end. Hence, by a theorem of Stallings, either G is finite or G splits nontrivially as an amalgamated free product or an HNN-extension where C is a finite group. Then are again finitely generated groups with context-free word-problem, and one can apply the entire preceding argument to them. Since G is finitely presentable and therefore accessible, the process of iterating this argument eventually terminates with finite groups, and produces a decomposition of G as the fundamental group of a finite graph-of-groups with finite vertex and edge groups. By a basic result of Bass–Serre theory it then follows that G is virtually free. The converse direction of the Muller–Schupp theorem is more straightforward. If G is a finitely generated virtually free group, then G admits a finite index normal subgroup N such that N is a finite rank free group. Muller and Schupp use this fact to directly verify that G has context-free word problem. Remarks and further developments The Muller–Schupp theorem is a far-reaching generalization of a 1971 theorem of Anisimov which states that for a finitely generated group G with a finite marked generating set X the word problem is a regular language if and only if the group G is finite. At the time the 1983 paper of Muller and Schupp was published, accessibility of finitely presented groups has not yet been established. Therefore, the original formulation of the Muller–Schupp theorem said that a finitely generated group is virtually free if and only if this group is accessible and has context-free word problem. A 1985 paper of Dunwoody proved that all finitely presented groups are accessible. Since finitely generated groups with context-free word problem are finitely presentable, Dunwoody's result together with