2025/07/28 by Gaifullin, Alexander A.
#11H56 #20E05 #20F18 #20F28 #20F34 (Primary) #57M07 (Secondary) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2507.20710
Let Σgb be a compact oriented surface of genus g with b boundary components, where b∈\0,1\. The Johnson kernel Kgb is the subgroup of the mapping class group Mod(Σgb) generated by Dehn twists about separating simple closed curves. Let Fn be a free group with n generators. The Torelli group for Out(Fn) is the subgroup IOn\subsetOut(Fn) consisting of all outer automorphisms that act trivially on the abelianization of Fn. Long standing questions are whether the groups Kgb and [IOn,IOn] or their abelianizations (Kgb)ab and [IOn,IOn]ab are finitely generated for g≥3 (respectively, n≥3). During the last 15 years, these questions were answered positively for g≥4 and n≥4, respectively. Nevertheless, the cases of g=3 and n=3 remained completely unsettled. In this paper, we prove that the abelianizations (K3b)ab and [IO3,IO3]ab are finitely generated. Our approach is based on a new general sufficient condition for a module over a Laurent polynomial ring to be finitely generated as an abelian group.