Recently I have been studing Mapping class group I know the mapping class group of genus 1 surface is $SL(2,Z)$ the a natural question is "What is the mapping class group of a genus three surface? "
Thanks
2026-03-28 03:51:40.1774669900
Mapping class group.
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Theorem 1 of "A SIMPLE PRESENTATION FOR THE MAPPING CLASS GROUP OF AN ORIENTABLE SURFACE " BRONISLAW WAJNRYB. ISRAEL JOURNAL OF MATHEMATICS, Vol. 45, Nos. 2-3, 1983. Gives a presentation of the mapping class group of a closed orientable surface.
In the case of genus 3 the group is generated by 7 Dehn twists along certain curves on the surface. Wajnryb writes down a complete set of relations (which are quite long so I won't copy them here).