##
**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.

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 |