Linearity of some low-complexity mapping class groups

Abstract

By analyzing known presentations of the pure mapping groups of orientable surfaces of genus g with b boundary components and n punctures in the cases when {g=0} with b and n arbitrary, and when {g=1} and {b+n} is at most 3, we show that these groups are isomorphic to some groups related to the braid groups and the Artin group of type {D_{4}}. As a corollary, we conclude that the pure mapping class groups are linear in these cases.

Artin groups of types $F_4$ and $H_4$ are not commensurable with that of type $D_4$

“…It is known that the pure mapping class group of the three times punctured torus is isomorphic to A[D 4 ], the Artin group of type D 4 modulo its center. This fact is implicit in the description of the general presentations for the mapping class groups of punctured surfaces given by Labruère and Paris in [13], and can also be deduced from the Gervais presentation [10], as it was shown in [20]. We will use it to describe the abstract commensurator of A[D 4 ] in the following theorem.…”

“…The fact that PM(Σ 1,0 , P 3 ) is isomorphic to A[D 4 ] with the given presentation on Dehn twist generators as the standard Artin generators for A[D 4 ] was shown in [20,Corollary 9]. It follows from Korkmaz's work [12] that Comm(M * (Σ 1,0 , P 3 )) is isomorphic to M * (Σ 1,0 , P 3 ), see [1,Theorem 3.1].…”

“…Our approach is similar: we use a realization of A[D 4 ] as the pure mapping class group PM(Σ 1,0 , P 3 ) of the torus with three punctures [20] (a fact which deserves to be more broadly known in the mathematical community). This yields a description of the commensurator Comm(A[D 4 ]) of A[D 4 ] as the extended mapping class group M * (Σ 1,0 , P 3 ) in Theorem 2.…”