Dehn twists on nonorientable surfaces

Abstract

Let a and b be two simple closed curves on an orientable surface S such that their geometric intersection number is greater than 1. The group generated by corresponding Dehn twists ta and t b is known to be isomorphic to the free group of rank 2. In this paper we extend this result to the case of a nonorientable surface.

Roots of crosscap slides and crosscap transpositions

“…and Proposition 4.4 of [13] implies that k 2 = 0 and (2m 1 + 2m 2 + 1)(2k + 1) = 1 which is a contradiction.…”

Crosscap slides and the level 2 mapping class group of a nonorientable surface

“…It follows that F commutes with Y i , T α i , T β j , and thus by Theorem 3.3 it belongs to the center of M(N g ) which is trivial, according to [18,Corollary 6.3]. Hence F = 1 and Let N g−1 be the surface obtained by replacing the leftmost crosscap in Fig.…”

Commensurability of geometric subgroups of mapping class groups

“…For M = M −2 the statement is trivial, hence assume that M = M −2 . By Proposition 3.4 of [13] and Lemma 4.1 of [16] there exists a circle b, generic in N, such that I(a, b) > 0. In particular b is not a boundary circle of N, hence by Proposition 2.2 and Theorem 3.6, the twist t b is nontrivial in M, that is b is generic in M.…”

“…The mapping class group of a Klein bottle is generated by a twist and a crosscap slide [9], and is isomorphic to Z 2 × Z 2 . The description of mapping class groups of a Klein bottle with one puncture and a Klein bottle with one hole can be found in the appendix to [16]. In particular, we will use the following proposition.…”

“…It is known [13] that an orientable surface has a pantalon decomposition if and only if 2g + r + s > 2 and M = M 3 0 . As observed in Section 5 of [16], one needs to add two more "pieces" in order to decompose nonorientable surfaces, namely a Möbius strip with one puncture and a Möbius strip with an open disk removed. We call these surfaces a skirt of type I and II, respectively (cf Figure 2).…”