The R∞ property for nilpotent quotients of surface groups

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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, called the R∞-nilpotency degree of G, such that the group G/ γc₊₁(G) has the R∞ property, where γᵣ(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 Sg of genus g>1. For the fundamental group of the non-orientable surface Ng (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 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 Sg has R∞-solvability degree 2.