Baumslag-Solitar subgroups of the mapping class group

2026/07/16 by Pankaj Kapari, Dev Krishna, Kashyap Rajeevsarathy
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Abstract

For g≥ 2 and nonzero integers p,q, let Mod(S_g) be the mapping class group of a closed oriented surface S_g of genus g, and let BS(p,q) be the Baumslag-Solitar group. We provide necessary and sufficient conditions under which two mapping classes G,F∈ Mod(S_g) generate a subgroup isomorphic to BS(p,q). In particular, if BS(p,q) embeds in Mod(S_g), then |p|=|q| and G,F are reducible mapping classes of infinite order. We also construct subgroups isomorphic to BS(p,p) and BS(p,-p) for p>1. Finally, we show that every infinite metacyclic subgroup and every Baumslag-Solitar subgroup of Mod(S_g) lifts to an isomorphic subgroup of Diff⁺(S_g), that is, these subgroups satisfy the generalized Nielsen realization.

Related

• Perfect kernel of generalized Baumslag-Solitar groups
• The topology of Baumslag-Solitar representations
• Infinite metacyclic subgroups of the mapping class group
• The Nielsen Realization Problem for Non-Orientable Surfaces
• Commensurability of Baumslag-Solitar groups