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Dehn’s algorithm revisited, with applications to simple curves on surfaces. (English) Zbl 0624.20033

Combinatorial group theory and topology, Sel. Pap. Conf., Alta/Utah 1984, Ann. Math. Stud. 111, 451-478 (1987).
[For the entire collection see Zbl 0611.00010.]
Let \(\Gamma\) be a Fuchsian group acting in the hyperbolic plane \({\mathbb{H}}\) with quotient space M. Let R be a fundamental region for the action of \(\Gamma\) and let \({\mathcal T}\) be the associated tesselation. If \(\gamma\) is a geodesic on M, the geodesic word \(<\gamma >\) of \(\gamma\) is the word determined by the sequence of generators corresponding to the edges of \({\mathcal T}\) cut by a lift of \(\gamma\) to \({\mathbb{H}}\). The hyperbolic length of \(\gamma\) and the word length of \(<\gamma >\) in \(\Gamma\) (equal to the number of sides of \({\mathcal T}\) cut by \(\gamma)\) are always comparable, but a geodesic word is not in general shortest. However, with careful choice of a metric on M, determined by a geometrical condition on R, geodesic paths give shortest words in \(\Gamma\). Such a choice of a metric is always possible without changing the topology of M. The proof is entirely by studying the geometry of \({\mathcal T}\). One obtains precise conditions for determining shortest words in \(\Gamma\), equality of shortest words, and shortest words in conjugacy classes. This gives a counter example to an assertion of Zieschang, that cyclically shortest conjugate words are of the same length. However with a slight extra condition on the side pairings defining \(\Gamma\) the result is true.
These results are applied to give detailed necessary, though not sufficient, conditions for determining when a word in \(\Gamma\) is simple, that is, when the unique smooth closed geodesic in a given conjugacy class in \(\pi_ 1(M)\) is a simple curve on M. In particular, the problem for closed surfaces, which is complicated by the existence of a non- trivial relation in \(\Gamma\), is reduced to the case of a free group (surface without boundary), as discussed [in J. Lond. Math. Soc., II. Ser. 29, 331-342 (1984; Zbl 0507.57006)] by the same authors.

MSC:

20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
20F05 Generators, relations, and presentations of groups
57M99 General low-dimensional topology