We show that for any k ≥ 6 and g sufficiently large, the mapping class group of a surface of genus g can be generated by three elements of order k. We also show that this can be done with four elements of order 5. We additionally prove similar results for some permutation groups, linear groups, and automorphism groups of free groups.2000 Mathematics Subject Classification. Primary: 20F65; Secondary: 57M07, 20F05.

We provide a simple criterion for an element of the mapping class group of a closed surface to have normal closure equal to the whole mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator for the mapping class group when the genus is at least 3. We also give many examples of pseudo-Anosov normal generators, answering a question of D. D. Long. In fact we show that every pseudo-Anosov mapping class with stretch factor less than √ 2 is a normal generator. Even more, we give pseudo-Anosov normal generators with arbitrarily large stretch factors and arbitrarily large translation lengths on the curve graph, disproving a conjecture of Ivanov.

In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show that no big mapping class group satisfies the strong Tits alternative enjoyed by finite-type mapping class groups. We also give examples of big mapping class groups that fail to satisfy even the classical Tits alternative; consequently, these examples are not linear. Let S be a connected orientable surface without boundary that is of infinite type, so that π 1 (S) is infinitely generated. The mapping class group Map(S) is the group of homotopy classes of orientation-preserving homeomorphisms of S. The mapping class group of an infinite-type surface is often called a big mapping class group. Similarly, let S g be the compact connected orientable surface of genus g and let Map(S g) be its mapping class group. In this note we address several questions about subgroups of Map(S). We first prove some results about the triviality of centers of subgroups of Map(S). We then show that Map(S) never satisfies the strong Tits alternative enjoyed by Map(S g), as well as some related results. Centers. The center of Map(S g) is trivial for g ≥ 3, while its center is a cyclic group generated by the hyperelliptic involution when g = 1 or 2 ([4, Section 3.4]). The centers of mapping class groups of finite-type surfaces with punctures and boundary components were computed by Paris-Rolfsen ([12, Theorem 5.6]). Although Dehn twists about boundary components are always

We give a simple geometric algorithm that can be used to determine whether or not a post-critically finite topological polynomial is Thurston equivalent to a polynomial. If it is, the algorithm produces the Hubbard tree for the polynomial, hence determining the polynomial. If it is not, the algorithm produces a Levy cycle, certifying that the map is not equivalent to a polynomial. Our methods are rooted in geometric group theory: we consider a lifting map on a simplicial complex of isotopy classes of trees. As an application, we give a self-contained solution to Hubbard's twisted rabbit problem, which was originally solved by Bartholdi-Nekrashevych using iterated monodromy groups. We also state and solve a generalization of the twisted rabbit problem to the case where the number of post-critical points is arbitrarily large.

We provide a simple criterion for an element of the mapping class group of a closed surface to be a normal generator for the mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator for the mapping class group when the genus is at least 3. We also give many examples of pseudo-Anosov normal generators, answering a question of D. D. Long. In fact, we show that every pseudo-Anosov mapping class with stretch factor less than p 2 is a normal generator. Even more, we give pseudo-Anosov normal generators with arbitrarily large stretch factors and arbitrarily large translation lengths on the curve graph, disproving a conjecture of Ivanov.