[Paper Review] Twisted conjugacy classes in nilpotent groups

В. А. Романьков|arXiv (Cornell University)|Mar 20, 2009

Geometric and Algebraic Topology8 references3 citations

TL;DR

This paper establishes an algorithm to compute the Reidemeister number $ R(\varphi) $ for any automorphism $ \varphi $ of a finitely generated nilpotent group. It proves that free nilpotent groups of rank $ r=2 $ or $ 3 $ with class $ c \geq 4r $, or rank $ r \geq 4 $ with class $ c \geq 2r $, belong to the $ R_\infty $ class—meaning all their automorphisms have infinite Reidemeister numbers, a key result in twisted conjugacy theory.

ABSTRACT

Let $N$ be a finitely generated nilpotent group. Algorithm is constructed such, that for every automorphism $ϕ\in Aut(N)$ defines the Reidemeister number $R(ϕ).$ It is proved that any free nilpotent group of rank $r = 2$ or $r = 3$ and class $c \geq 4r,$ or rank $r \geq 4$ and class $c \geq 2r,$ belongs to the class $R_{\infty}.$

Motivation & Objective

• To develop an effective algorithm for computing the Reidemeister number $ R(\varphi) $ for any automorphism $ \varphi $ of a finitely generated nilpotent group.
• To determine for which free nilpotent groups all automorphisms have infinite Reidemeister numbers, i.e., belong to the $ R_\infty $ class.
• To extend the understanding of twisted conjugacy in nilpotent groups by analyzing the structure of $ \varphi $-conjugacy classes via central series and induced automorphisms.
• To establish sufficient conditions on rank and class for free nilpotent groups to lie in $ R_\infty $, based on fixed-point behavior of induced automorphisms on abelian quotients.

Proposed method

• Uses a recursive algorithm based on the upper central series of a finitely generated nilpotent group, reducing the problem to abelian quotients.
• Applies Lemma 2.4 to lift $ \bar{\varphi} $-conjugacy classes from $ N/C_{k} $ to $ N $, using the subgroup $ L(C_k, \varphi_g) $ to determine the number of preimage classes.
• Employs the identity $ R(\varphi) = [C : L(C, \varphi)] $ for central subgroups $ C $, where $ L(C, \varphi) = \{ c \in C \mid \exists x \in G : x\varphi = c x \} $, to compute indices.
• Uses induction on the nilpotency class, assuming the algorithm works for groups of class $ \leq k-1 $, and extends it to class $ k $.
• Applies Formanek's classification of automorphism-invariant elements in free nilpotent groups to identify cases where $ \mathrm{Fix}_{\bar{\varphi}_i}(A_i) \neq 1 $, implying $ R(\varphi) = \infty $.
• Relies on the fact that if $ \mathrm{Fix}_{\varphi_i}(N_i) \neq 1 $, then $ R(\varphi_i) = \infty $, which propagates to $ R(\varphi) = \infty $ via the induced automorphism structure.

Experimental results

Research questions

• RQ1 For which finitely generated nilpotent groups does every automorphism have infinite Reidemeister number?
• RQ2 Can an effective algorithm be constructed to compute $ R(\varphi) $ for any automorphism $ \varphi $ of a finitely generated nilpotent group?
• RQ3 Under what conditions on rank $ r $ and class $ c $ do free nilpotent groups $ N_{rc} $ belong to the $ R_\infty $ class?
• RQ4 How does the fixed-point behavior of induced automorphisms on abelian quotients $ A_i = \zeta_{i+1}N / \zeta_i N $ determine the Reidemeister number?
• RQ5 What is the role of the subgroup $ L(C, \varphi) $ in determining the number of $ \varphi $-conjugacy classes in central extensions?

Key findings

• An effective algorithm is constructed to compute $ R(\varphi) $ for any automorphism $ \varphi $ of a finitely generated nilpotent group.
• For free nilpotent groups of rank $ r=2 $ or $ 3 $, if the class $ c \geq 4r $, then $ N_{rc} \in R_\infty $.
• For free nilpotent groups of rank $ r \geq 4 $, if the class $ c \geq 2r $, then $ N_{rc} \in R_\infty $.
• The key mechanism is the existence of a nontrivial fixed-point subgroup $ \mathrm{Fix}_{\bar{\varphi}_i}(A_i) \neq 1 $ in some abelian quotient $ A_i $, which implies $ R(\varphi) = \infty $.
• The result relies on Formanek's classification of automorphism-invariant elements in free nilpotent groups, which identifies the critical rank-class thresholds.
• The proof uses induction on the nilpotency class and leverages the structure of central series and induced automorphisms to lift conjugacy class information from quotients.