TAILIEUCHUNG - On the linearity of certain mapping class groups

S. Bigelow proved that the braid groups are linear. That is, there is a faithful representation of the braid group into the general linear group of some field. Using this, we deduce from previously known results that the mapping class group of a sphere with punctures and hyperelliptic mapping class groups are linear. In particular, the mapping class group of a closed orientable surface of genus 2 is linear. | Turk J Math 24 (2000) , 367 – 371. ¨ ITAK ˙ c TUB On the Linearity of Certain Mapping Class Groups Mustafa Korkmaz Abstract S. Bigelow proved that the braid groups are linear. That is, there is a faithful representation of the braid group into the general linear group of some field. Using this, we deduce from previously known results that the mapping class group of a sphere with punctures and hyperelliptic mapping class groups are linear. In particular, the mapping class group of a closed orientable surface of genus 2 is linear. Key Words: Mapping class groups, Braid groups, Linear groups Introduction One of the well-known open problem in the theory of mapping class groups is that whether these groups are linear or not (cf. [2], Problem 30, p. 220). A group is called linear if it has a faithful representation into GL(n, F ) for some field F and for some integer n. Recently, S. Bigelow [1] proved that the braid groups are linear. The braid group Bn on n strings divided out by its center is isomorphic to a finite index subgroup of the mapping class group of a sphere with n+1 marked points. Using this, we observe that the mapping class group of a sphere with marked points and that the hyperelliptic mapping class groups, which are defined below, are linear. In particular, the mapping class group of a closed orientable surface of genus 2 is linear. The linearity of the mapping class group of a surface of genus ≥ 3 still remains open. Preliminaries We first set up the notations and state the theorems used in the proof of the results of this paper. Then we prove our results. 1991 AMS Subject Classification Primary 57M60, 57N05; Secondary 20F34, 20F36, 30F99 367 KORKMAZ Let S be a compact connected orientable surface of genus g with r marked points (also called punctures) contained in the interior of S and with s boundary components. The mapping class group Msg,r of S is defined to be the group of isotopy classes of orientation preserving diffeomorphisms of S which

• Toán học
• Mapping class groups
• Braid groups
• Linear groups
• Orientable surface
• Hyperelliptic mapping
• Conjugacy classes
• Hyperelliptic mapping class group
• Finite subgroups of mapping class group
• Algebraic methods
• Fundamental set for the action of Zn
• Injective simplicial maps
• Arc complex
• Extended mapping class group
• Asymptotic faithfulness
• subjects of mathematics
• science reports
• studied mathematics
• natural sciences
• scientific research
• Dimension
• Braid relation
• Chain relation
• Generalized Matsumoto relation
• Closed orientable surface