Dehn twists have roots

Dehn twists have roots

Dan Margalit  and  Saul Schleimer Dan Margalit
Department of Mathematics
503 Boston Ave
Tufts University
Medford, MA 02155
dan.margalit@tufts.edu Saul Schleimer
Department of Mathematics
University of Warwick
Coventry, CV4 7AL, UK
s.schleimer@warwick.ac.uk
July 20, 2019
This work is in the public domain.

Let denote a closed, connected, orientable surface of genus , and let denote its mapping class group, that is, the group of homotopy classes of orientation preserving homeomorphisms of .

Fact. If , then every Dehn twist in has a nontrivial root.

It follows from the classification of elements in that Dehn twists are primitive in the mapping class group of the torus.

For Dehn twists about separating curves, the fact is well-known: if is a separating curve then a square root of the Dehn twist is obtained by rotating the subsurface of on one side of through an angle of . In the case of nonseparating curves, the issue is more subtle. We give two (equivalent) constructions of roots below.

Geometric construction.

Fix . Let be a regular -gon. Glue opposite sides to obtain a surface . The rotation of about its center through angle induces a periodic map of . Notice that fixes the points that are the images of the vertices of . Let be the surface obtained from by removing small open disks centered at and . Define .

Let and be annular neighborhoods of the boundary components of . Modify by an isotopy supported in so that

  • is the identity,

  • is a –left Dehn twist, and

  • is a –right Dehn twist.

Identify the two components of to obtain a surface and let be the induced map. Then is a left Dehn twist along the gluing curve, which is nonseparating.

Algebraic construction.

Let be curves in where intersects once for each , and all other pairs of curves are disjoint. If is odd, then a regular neighborhood of has two boundary components, say, and , and we have a relation in as follows:

This relation comes from the Artin group of type , in particular, the factorization of the central element in terms of standard generators [2]. In the case , the curves and are isotopic nonseparating curves; call this isotopy class . Using the fact that commutes with each , we see that

Other roots.

All roots of Dehn twists are obtained in a similar way. That is, if is a root of a Dehn twist then the canonical reduction system for is [1]. By the Nielsen–Thurston classification for surface homeomorphisms [3], if we cut the surface along , then restricts to a finite order element.

Roots of half-twists.

Let be the sphere with punctures (or cone points) and let be a curve in with 2 punctures on one side and on the other. On the side of with 2 punctures, we perform a left half-twist, and on the other side we perform a –right Dehn twist by arranging the punctures so that one puncture is in the middle, and the other punctures rotate around this central puncture. The power of the composition is a left half-twist about . Thus, we have roots of half-twists in for . There is a 2-fold orbifold covering where the relation from our algebraic construction above descends to this relation in . A slight generalization of this construction gives roots of half-twists in any with .

Roots of elementary matrices.

If we consider the map given by the action of on , we also see that elementary matrices in have roots; for instance, we have

By stabilizing, we obtain cube roots of elementary matrices in for .

Roots of Nielsen transformations.

Let denote the free group generated by , let denote the group of automorphisms of , and assume . A Nielsen transformation in is an element conjugate to the one given by and for . The following automorphism is the square root of a Nielsen transformation in for .

Taking quotients, this gives a square root of a Nielsen transformation in and, multiplying by , a square root of an elementary matrix in , . Finally, our roots of Dehn twists in can be modified to work for punctured surfaces, thus giving “geometric” roots of Nielsen transformations in .

References

  • [1] Joan S. Birman, Alex Lubotzky, and John McCarthy. Abelian and solvable subgroups of the mapping class groups. Duke Math. J., 50(4):1107–1120, 1983.
  • [2] Egbert Brieskorn and Kyoji Saito. Artin-Gruppen und Coxeter-Gruppen. Invent. Math., 17:245–271, 1972.
  • [3] William P. Thurston. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.), 19(2):417–431, 1988.
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