Type of Document	Dissertation
Author	Mortada, Jamil W.
URN	etd-04022011-121106
Title	Artin and Dehn twist subgroups of the mapping class group
Degree	Doctor of Philosophy
Department	Mathematics, Department of
Advisory Committee	Advisor Name Title Sergio Fenley Committee Chair Eriko Hironaka Committee Co-Chair Eric Klassen Committee Member Philip Bowers Committee Member Wolfgang Heil Committee Member Jorge Piekarewicz University Representative
Keywords	Artin relation Dehn twist Artin group Mapping class group
Date of Defense	2011-03-28
Availability	unrestricted
Abstract This dissertation investigates two types of subgroups in the mapping class group of an orientable surface. The first type of subgroups are isomorphic images of Artin groups. The second type of subgroups is one which is generated by three Dehn twists along simple closed curves with small geometric intersections. Let S be a compact orientable surface. The mapping class group, Mod(S), of S is the group of isotopy classes of orientation preserving homeomorphisms of S fixing the boundary pointwise. Mod(S) is a very rich and complex object. In this dissertation, we make progress toward understanding the structure of the above mentioned subgroups of Mod(S). We tackle three problems. The first problem focuses on finding embeddings of Artin groups into Mod(S). The second problem involves finding Artin relations of every length in Mod(S). And the third problem deals with understanding subgroups of Mod(S) generated by three Dehn twists along curves with small geometric intersections. While it is easy to find nontrivial homomorphisms of Artin groups into Mod(S), the question of whether such homomorphisms are injective is quite hard. In this dissertation, we find embeddings of the Artin groups A(Bn), A(H3), A(I2(n)), and most notably A(widetilde{A}{n-1}) into Mod(S). Further, we prove that if a collection a_1,...,a_n is a collection of simple closed curves in S has curve graph widetilde{A}{n-1} and N is a closed regular neighborhood of the union of the a_i, then the subgroup of Mod(N) generated by the (left) Dehn twists T_i along a_i is isomorphic to A(widetilde{A}{n-1}) almost all the time. In the second problem, we study Artin relations in the mapping class group. If l is an integer bigger or equal to 2, then a and b satisfy the Artin relation of length l if aba... = bab..., where each side of the equality has l terms. We give explicit elements of Mod(S) satisfying Artin relations of every integer length l bigger or equal to 2. By direct computations, we find elements x and y in Mod(S) satisfying Artin relations of every even length bigger or equal to 8 and every odd length bigger or equal 3. Then using the theory of Artin groups, we give two methods for finding Artin relations in Mod(S). The first yields Artin relations of every length bigger or equal to 3, while the second provides Artin relations of every even length bigger or equal 6. In the last two cases, we also show that x and y generate the Artin group A(I2(l)), where l is the length of the Artin relation satised by x and y. The third problem is concerned with understanding subgroups in Mod(S) generated by three Dehn twists along curves with small geometric intersections. Let a_1, a_2, and a_3 be distinct isotopy classes of essential simple closed curves in an orientable surface S. Assume that i(a_j ; a_k) is less than 3 for all j; k. Denote by T_i the (left) Dehn twist along a_i, and let G represent the subgroup of Mod(S) generated by T_1, T_2, and T_3. Set (x_12; x_13; x_23) = (i(a_1; a_2); i(a_1; a_3); i(a_2; a_3)). We find explicit presentations for G when (x_12; x_13; x_23) = (0; 0; 0), (1; 0; 0), (2; 0; 0), (1; 0; 1), and (1; 1; 1). For the triple (2; 1; 0), there are two cases to consider (see subsections 7.8.1 and 7.8.2). In both cases, we are not able to find an explicit presentation for G. Nevertheless, we prove that G is a subgroup of some Artin group A. Moreover, using the computer algebra software Magma, we show that G is finitely presented and is isomorphic to a subgroup of infinite index in A. Although we have obtained similar partial results for the triples (2; 2; 0), (2; 1; 0), (2; 1; 1), (2; 2; 0), and (2; 2; 2), we do not include them in this dissertation. While the three problems discussed above are seemingly disconnected, they are in fact intimately related. They reflect a beautiful interplay between Artin groups and mapping class groups.
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