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/γ c+1(G) has the R∞ property, where γr(G) is the rth 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, 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. 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.

Twisted conjugacy classes in unitriangular groups

“…The following result which gives the discription of an arbitrary automorphism of UT n (R) is a consequence of [11,Theorems 1,2,3] in assumption that R is an integral domain of zero characteristic.…”

“…We refer to the paper [5] for an overview of the families of groups which have been studied in this context until 2016. More recent results can be found in [2,13,19,20]. The author studied twisted conjugacy classes and the R ∞ -property for classical linear groups [5,[14][15][16][17].…”

Chevalley groups of types $B_n$, $C_n$, $D_n$ over certain fields do not possess the $R_{\infty}$-property

“…We refer to the paper [5] for an overview of the families of groups which have been studied in this context until 2016. More recent results can be found in [3,9,15,[17][18][19]. For the immediate consequences of the R ∞ -property for topological fixed point theory see [7].…”

The 𝑅∞-property for nilpotent quotients of Baumslag–Solitar groups

“…It now follows that the induced mapψ on any quotient BS c (m, m) has −1 as an eigenvalue and hence R(ψ) < ∞ ( See [3]). It follows that the R ∞ -nilpotency index of BS(m, m) is ∞.…”

“…Several families of groups have been studied by many authors. A non-exhaustive list of references is [1,2,3,4,5,6,7,8,9,10,14].…”