Conjecture on normal right-angled Artin subgroups of mapping class groups

Let S=Sg,pS=S_{g,p} be an orientable surface, and let Mod(S)\operatorname{Mod}(S) denote its mapping class group. Let NN be a nontrivial normal subgroup of Mod(S)\operatorname{Mod}(S) that is isomorphic to a right-angled Artin group. The groups occurring in the authors' construction include the free products

F,(F×Z),(F×F),(F×F×Z).F_\infty,\qquad \mathop{*}_{\infty}\left(F_\infty\times\mathbb{Z}\right),\qquad \mathop{*}_{\infty}\left(F_\infty\times F_\infty\right),\qquad \mathop{*}_{\infty}\left(F_\infty\times F_\infty\times\mathbb{Z}\right).

Normal RAAG conjecture. The subgroup NN is isomorphic to one of the right-angled Artin subgroups of Mod(S)\operatorname{Mod}(S) afforded by the authors' construction. In particular, NN is isomorphic to a free product of the groups

F,(F×Z),(F×F),(F×F×Z).F_\infty,\qquad \mathop{*}_{\infty}\left(F_\infty\times\mathbb{Z}\right),\qquad \mathop{*}_{\infty}\left(F_\infty\times F_\infty\right),\qquad \mathop{*}_{\infty}\left(F_\infty\times F_\infty\times\mathbb{Z}\right).

The construction gives examples of normal, non-free, right-angled Artin subgroups, while results for mapping classes with sufficiently small support exclude such groups in that setting. The conjecture seeks to classify all nontrivial normal right-angled Artin subgroups of mapping class groups and remains open.

Sources & referencesView supporting material

Primary source

Matt Clay, Johanna Mangahas and Dan Margalit, “Right-angled Artin groups as normal subgroups of mapping class groups”, arXiv:2001.10587 (2021).

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