The mapping class group of a genus two surface is linear. (English) Zbl 0999.57020

A group is called linear if there is a faithful representation i.e., an injective group homomorphism into the group of invertible matrices over some commutative ring. S. J. Bigelow [J. Am. Math. Soc. 14, No. 2, 471-486 (2001; Zbl 0988.20021)] and independently D. Krammer [Braid groups are linear, Ann. Math (2) 155, No. 1, 131-156 (2002)] showed that a representation of the braid groups constructed by Lawrence is faithful. The braid group on \(k\) strands can be defined as the mapping class group of a \(k\)-punctured disk. The question arises whether other mapping class groups are linear.
The authors show that the mapping class groups of punctured spheres are linear. These groups are quotients of braid groups by a central \(Z\). The representations for the latter is easily adapted.
The hyperelliptic mapping class group of a closed genus \(g\) surface is an extension of the mapping class group of the sphere with \(2g + 2\) punctures by the hyperelliptic involution. The involution can be detected by the standard symplectic representation, and hence hyperelliptic mapping class groups are linear. For genus 2, as is well known, the hyperelliptic mapping class group is the full mapping class group. Hence the result of the title.
The results and methods of the paper are the same as those in [M. Korkmaz, Turk. J. Math. 24, No. 4, 367-371 (2000; Zbl 0965.57013)], albeit there is an improvement on the dimensions of the representations.


57M99 General low-dimensional topology
20F38 Other groups related to topology or analysis
20F36 Braid groups; Artin groups
57M07 Topological methods in group theory
20C15 Ordinary representations and characters
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