The R_∞ property for nilpotent quotients of surface groups

2015/05/29 by Karel Dekimpe, Dekimpe, Karel, Daciberg Lima Gonçalves +1
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1505.07974

Abstract

It is well known that when G is the fundamental group of a closed surface of negative Euler characteristic, it has the R_∞ property. In this work we compute the least integer c, \it called the R_∞-nilpotency degree of G, such that the group G/ γ_c+1(G) has the R_∞ property, where γ_r(G) is the r-th term of the lower central series of G. We show that c=4 for G the fundamental group of any orientable closed surface S_g of genus g>1. For the fundamental group of the non-orientable surface N_g (the connected sum of g projective planes) this number is 2(g-1) (when g>2). A similar concept is introduced using the derived series G^(r) of a group G. Namely \it the R_∞-solvability degree of G, which is the least integer c such that the group G/G^(c) has the R_∞ property. We show that the fundamental group of an orientable closed surface S_g has R_∞-solvability degree 2.

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