The R ∞ property for nilpotent quotients of surface groups

A group G is said to have the R ∞ property if, for any automorphism φ of G , the number R ( φ ) of twisted conjugacy classes (or Reidemeister classes) is infinite. 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 theR ∞ ‐ nilpotency degree of G , such that the group G / γ c + 1( G )has the R ∞ property, whereγ 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, theR ∞ ‐ 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. As a by‐product of our research, we find a lot of new examples of nilmanifolds on which every self‐homotopy equivalence can be deformed into a fixed point free map.