Virtually Torsion-Free Covers of Minimax Groups
Virtually torsion-free covers of minimax groups Peter Kropholler Karl Lorensen October 12, 2018 Abstract We prove that every finitely generated, virtually solvable minimax group can be expressed as a homomorphic image of a virtually torsion-free, virtually solvable minimax group. This result enables us to generalize a theorem of Ch. Pittet and L. Saloff-Coste about random walks on finitely generated, virtually solvable minimax groups. Moreover, the paper identifies properties, such as the derived length and the nilpotency class of the Fitting subgroup, that are preserved in the covering process. Finally, we determine exactly which infinitely generated, virtually solvable minimax groups also possess this type of cover. Couvertures virtuellement sans torsion de groupes minimax R´esum´e Nous prouvons que tout groupe de type fini minimax et virtuellement r´esoluble peut ˆetre exprim´ecomme l’image homomorphe d’un groupe minimax virtuellement r´esoluble et virtuellement sans torsion. Ce r´esultat permet de g´en´eraliser un th´eor`eme de Ch. Pittet et L. Saloff-Coste concernant les marches al´eatoires sur les groupes de type fini minimax et virtuellement r´esolubles. En outre, l’article identifie des propri´et´es con- serv´ees dans le processus de couverture, telles que la classe de r´esolubilit´eet la classe de nilpotence du sous-groupe de Fitting. Enfin, nous d´eterminons exactement quels groupes de type infini minimax et virtuellement r´esolubles admettent ´egalement ce type de couverture. Mathematics Subject Classification (2010): 20F16, 20J05 (Primary); 20P05, 22D05, 60G50, 60B15 (Secondary). arXiv:1510.07583v3 [math.GR] 11 Oct 2018 Keywords: virtually solvable group of finite rank, virtually solvable minimax group, random walks on groups, groupe virtuellement r´esoluble de rang fini, groupe minimax virtuellement r´esoluble, marches al´eatoires sur les groupes.
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