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The R1 and S1 Properties for Linear Algebraic Groups

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J. Group Theory 19 (2016), 901–921 DOI 10.1515/jgth-2016-0004 © de Gruyter 2016 The R and S properties for 1 1 linear algebraic groups Alexander Fel’shtyn and Timur Nasybullov Communicated by Evgenii I. Khukhro Abstract. In this paper we study twisted conjugacy classes and isogredience classes for automorphisms of reductive linear algebraic groups. We show that reductive linear alge- braic groups over some fields of zero characteristic possess the R and S properties. 1 1 1 Introduction Let ' G G be an endomorphism of a group G. Then two elements x; y of W ! G are said to be twisted '-conjugate if there exists a third element z G such 2 that x zy'.z/ 1. The equivalence classes of this relation are called the twisted D conjugacy classes or the Reidemeister classes of '. The Reidemeister number of ' denoted by R.'/, is the number of twisted conjugacy classes of '. This number is either a positive integer or and we do not distinguish different infi- 1 nite cardinal numbers. An infinite group G possesses the R -property if for every automorphism ' of G the Reidemeister number of ' is infinite.1 The interest in twisted conjugacy relations has its origins, in particular, in Nielsen–Reidemeister fixed point theory (see, e.g. [7, 34]), in Arthur–Selberg the- ory (see, e.g. [1,50]), in algebraic geometry (see, e.g. [28]), in Galois cohomology [49] and in the theory of linear algebraic groups (see, e.g. [52]). In representation theory twisted conjugacy probably occurs first in Gantmacher’s paper [20] (see, e.g. [45, 51]). The problem of determining which classes of discrete infinite groups have the R -property is an area of active research initiated by Fel’shtyn and Hill in 1994 (see1 [13]). Later, it was shown by various authors that the following groups have the R -property: non-elementary Gromov hyperbolic groups (see [8, 38]); rela- tively1 hyperbolic groups (see [9]); Baumslag–Solitar groups BS.m; n/ except for BS.1; 1/ (see [10]), generalized Baumslag–Solitar groups, that is, finitely gener- ated groups which act on a tree with all edge and vertex stabilizers infinite cyclic (see [37]); the solvable generalization of BS.1; n/ given by the short exact sequence 1 ZŒ 1 Zk 1 (see [53]); a wide class of saturated weakly ! n ! ! ! The second author is supported by Russian Science Foundation (project 14-21-00065). 902 A. Fel’shtyn and T. Nasybullov branch groups (including the Grigorchuk group (see [27]) and the Gupta–Sidki group (see [29])) (see [15]), Thompson’s groups F (see [2]) and T (see[3, 22]); the generalized Thompson groups Fn; 0 and their finite direct products (see [21]); Houghton’s groups (see [23,35]); symplectic groups Sp.2n; Z/, the mapping class groups ModS of a compact oriented surface S with genus g and p boundary com- ponents, 3g p 4 > 0, and the full braid groups Bn.S/ on n > 3 strands of C a compact surface S in the cases where S is either the compact disk D, or the sphere S 2 (see [11]); some classes of Artin groups of infinite type (see [36]); exten- sions of SL.n; Z/, PSL.n; Z/, GL.n; Z/, PGL.n; Z/, Sp.2n; Z/, PSp.2n; Z/, for n > 2, by a countable abelian group, and normal subgroups of SL.n; Z/, n > 2, not contained in the center (see [40]); GL.n; K/, SL.n; K/, Sp.2n; K/ and On.K/ if n > 2 and K is an infinite integral domain which has zero characteristic and for which Aut.K/ is periodic (see [42,44]); Chevalley groups over fields of zero char- acteristic which have finite transcendence degree over Q (see [43]); irreducible lattices in a connected semisimple Lie group G with finite center and real rank at least 2 (see [41]); non-amenable, finitely generated residually finite groups (see [17]) (this class gives a lot of new examples of groups with the R -property); n Ì 1 n Ì 1 some metabelian groups of the form Q Z and ZŒ p Z (see [12]); lamp- lighter groups Zn Z if and only if 2 n or 3 n (see [25]); free nilpotent groups Nrc o j j of rank r 2 and class c 9 (see [26]), Nrc of rank r 2 or r 3 and class D D D c 4r; or rank r 4 and class c 2r; any group N2c for c 4, every free solv- able group S2t of rank 2 and class t 2, any free solvable group Srt of rank r 2 and class t big enough (see [47]); some crystallographic groups (see [6,39]). Recently, in [5] it was proven that Nrc, r > 1 possesses the R -property if and only if c 2r. 1 Let ‰ be an element of Out.G/ Aut.G/=Inn.G/ (we consider ‰ as a col- D lection of ordinary automorphisms a Aut.G/). We say that two automorphisms 2 a; b ‰ are similar (or isogredient) if b ' a' 1 for some h G, where 2 D h h 2 ' .g/ hgh 1 is an inner automorphism induced by the element h (see [38]). h D Let S.‰/ be the set of isogredience classes of automorphisms representing ‰. Denote by S.‰/ the cardinality of the set S.‰/. The group G is said to possess the S -property if for every ‰ the set S.‰/ is infinite, i.e. S.‰/ (see [18]). 1 D 1 In this paper we study the R and S properties for linear algebraic groups. The first results in this direction1 were1 obtained for some classes of Chevalley groups by Nasybullov in [43]. In Section 3 we extend the previous result from [43] and prove the following. Theorem 3.2. Let G be a Chevalley group of type ˆ over a field F of zero char- acteristic. If the transcendence degree of F over Q is finite, then G possesses the R -property. 1 The R and S properties for linear algebraic groups 903 1 1 The following main theorem is proved in Section 4. Theorem 4.1. Let F be an algebraically closed field of zero characteristic such that the transcendence degree of F over Q is finite. If a reductive linear algebraic group G over the field F has a nontrivial quotient group G=R.G/, where R.G/ is the radical of G, then G possesses the R -property. 1 These theorems cannot be generalized to groups over a field of non-zero char- acteristic, by the following theorem of Steinberg [52, Theorem 10.1]. Theorem. Let G be a connected linear algebraic group and let ' be an endomor- phism of G onto G. If ' has a finite number of fixed points, then G x'.x 1/ x G : D ¹ W 2 º Indeed, any semisimple linear algebraic group over an algebraically closed field of positive characteristic possesses an automorphism ' with finitely many fixed points (the Frobenius morphism, see [48, Section 3.2]), therefore, this group coin- 1 cides with the set x'.x / x G Œe', hence R.'/ 1 and such a group ¹ W 2 º D D cannot possess the R -property. Throughout the paper,1 by an automorphism of a linear algebraic group G we mean an automorphism of the abstract group underlying G (without any extra conditions). If T1;T2;::: are algebraically independent over Q, the fields Q, Q.T1;:::;Tk/ .k 1/ are algebraically closed fields of zero characteristic with the finite tran- scendence degree over Q. Then reductive linear algebraic groups over these fields possess the R -property. In Section1 5 we prove that an infinite reductive linear algebraic group G over a field F of zero characteristic and finite transcendence degree over Q which pos- sesses an automorphism ' with finite Reidemeister number is a torus. In Section 6 we prove the following theorem. Theorem 6.3. Let F be an algebraically closed field of zero characteristic such that the transcendence degree of F over Q is finite. If a reductive linear alge- braic group G over the field F has a nontrivial quotient group G=R.G/, then G possesses the S -property. 1 2 Preliminaries In this section we recall some preliminary statements which are used in the paper. A lot of the results we use are thoroughly presented in [43], this can be used as background material. 904 A. Fel’shtyn and T. Nasybullov The symbols In and On m indicate the identity n n matrix and the n m matrix with zero entries, respectively. If A is an n n matrix and B is an m m matrix, then the symbol A B denotes the direct sum of the matrices A and B, ˚ i.e. the block-diagonal .m n/ .m n/ matrix C C ! A On m : Om n B It is obvious that for a pair of n n matrices A1;A2 and for a pair of m m matrices B1;B2 we have 1 1 1 .A1 B1/.A2 B2/ A1A2 B1B2 and .A1 B1/ A B : ˚ ˚ D ˚ ˚ D 1 ˚ 1 The symbols G H and G H denote the direct product and the central prod- ı uct of the groups G and H, respectively. If g is an element of a group G, then 'g denotes the inner automorphism in- duced by the element g. The following lemma can be found in [16, Corollary 2.5]. Lemma 2.1. Let ' be an automorphism of a group G and let 'g be an inner auto- morphism of the group G.
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