Twisted conjugacy classes in unitriangular groups

Abstract

Let R be an integral domain of zero characteristic. In this note we study the Reidemeister spectrum of the group UT n (R) of unitriangular matrices over R. We prove that if R + is finitely generated and n > 2|R * |, then UT n (R) possesses the R ∞ -property, i. e. the Reidemeister spectrum of UT n (R) contains only ∞, however, if n ≤ |R * |, then the Reidemeister spectrum of UT n (R) has nonempty intersection with N. If R is a field, then we prove that the Reidemeister spectrum of UT n (R) coincides with {1, ∞}, i. e. in this case UT n (R) does not possess the R ∞ -property. The author is supported by the Research Foundation -Flanders (FWO), app. 12G0317N.R ∞ -property. The problem of classifying groups which possess the R ∞ -property was proposed by A. Fel'shtyn and R. Hill in [8]. The study of this problem has been quite an active research topic in recent years. 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]. For the immediate consequences of the R ∞ -property for topological fixed point theory see [9]. Some aspects of the R ∞ -property can be found in [7].The Reidemeister numbers for automorphisms of nilpotent groups are studied only for free nilpotent groups [1,18], nilpotent quotients of surface groups [2] and some very specific nilpotent groups motivated by geometry [9]. Denote by N r,c = F r /γ c+1 (F r ) the free nilpotent group of rank r and nilpotency class c. V. Roman'kov proved in [18] that if r ≥ 4 and c ≥ 2r, then the group N r,c possesses the R ∞ -property. He also found the Reidemeister spectrum for groups N 2,2 , N 2,3 , N 3,2 . K. Dekimpe and D. Gonçalves extended the result of Roman'kov in [1] proving that the group N r,c possesses the R ∞ -property if and only if r ≥ 2 and c ≥ 2r. The authors of [3] studied the Reidemeister spectrum for the groups N r,c for c < 2r. In this note we study the Reidemeister spectrum and the R ∞ -property for another class of nilpotent groups, namely for groups UT n (R) of upper unitriangular matrices over integral domains R of zero characteristic. The group UT n (Z) is very important for studying nilpotent groups in general since if G is a finitely generated torsion free nilpotent group, then it can be imbedded into UT n (Z) for an appropriate positive integer n. The group UT n (R) is both nilpotent and classical linear, so, the present work is a natural continuation of works [5,[14][15][16][17]. The main results of the paper are formulated in the following two theorems.

Twisted conjugacy in direct products of groups

“…Theorem 3] that certain reductive linear algebraic groups G have the R ∞ -property by proving it for the quotient group G/R(G), which splits as a direct product of Chevalley groups. The latter can also be proved by combining the results from [32,33,34,35,36] with Corollary 4.5.…”

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 author studied conditions which imply the R ∞ -property for different linear groups over rings [11,13,15] and fields [5,12,14,15]. In particular, it was proved that if F is a field of zero characteristic which has either finite transcendence degree over Q, or periodic group of automorphisms, then every Chevalley group (of normal type) over F possesses the R ∞ -property [12].…”

Twisted conjugacy in classical groups over certain domains of Characteristic $p>0$

“…Various other authors have studied R ∞ -property of linear groups. See [3], [13], [17], [18], [19], [20], [21]. Nasybullov [20] and Felshtyn-Nasybullov [3] proved that that a Chevally group of classical type over an algebraically closed field F of characteristic zero has the R ∞ -property if and only if F has finite transcendence degree over Q.…”