The R∞ property for nilpotent quotients of surface groups

Abstract

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 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. 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.