The R1 and S1 Properties for Linear Algebraic Groups

Home , Automorphism group, Hyperbolic group

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 algebraic 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 theory (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 generated 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]); extensions 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 characteristic 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 solvable 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 characteristic. 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 characteristic, by the following theorem of Steinberg [52, Theorem 10.1].

Theorem. Let G be a connected linear algebraic group and let ' be an endomorphism 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 transcendence 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 possesses 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 algebraic 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 induced 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 automorphism of the group G. Then R.''g / R.'/. D The next lemma is proved in [40, Lemma 2.1].

Lemma 2.2. Let 1 N G A 1 ! ! ! ! be an exact sequence of groups. Suppose that N is a characteristic subgroup of G and that A possesses the R -property. Then G also possesses the R -property. 1 1 Here we prove a similar result for the S -property. 1 Lemma 2.3. Let 1 N G A 1 ! ! ! ! be an exact sequence of groups. Suppose that N is a characteristic subgroup of G and that A possesses the S -property. Then G also possesses the S -property. 1 1 Proof. Let ' be an automorphism of the group G. Since N is the characteristic subgroup of G, it follows that ' induces the automorphism ' of the group A. Since the group A has the S -property, there exists an inﬁnite set of elements 1 g ; g ;::: of the group A such that 'g ' and 'g ' are not isogredient for i j . 1 2 i j ¤ Suppose that S.'Inn.G// < . Then there exists a pair of isogredient auto- 1 morphisms in the set 'g1 ';'g2 ';::: . Suppose that 'gi ' and 'gj ' are isogredient for i j . Then for some element h G we have ¤ 2 1 'g ' ' 'g '' : i D h j h The R and S properties for linear algebraic groups 905 1 1

From this equality we have the following equality in the group Aut.A/ 1 'g ' ' 'g '' ; i D h j h this contradicts the choice of the elements g1; g2;::: . Let be a map from the set of rational numbers Q to the set 2 of all subsets of the set of prime numbers , which acts on the irreducible fraction x a=b by D the rule .x/ all prime divisors of a all prime divisors of b : D ¹ º [ ¹ º The proof of the following lemma is presented in [43, Lemma 5].

Lemma 2.4. Let F be a field of zero characteristic and let x1; : : : ; xk be elements of F which are algebraically independent over the field Q. Let xk 1 be an element C of F such that the elements x1; : : : ; xk 1 are algebraically dependent over Q. Let ı be an automorphism of the field F Cwhich acts on these elements by the rule

ı xi t0ti xi ; i 1; : : : ; k 1; W 7! D C where t0; : : : ; tk 1 Q and t1; : : : ; tk 1 are not equal to 1. If .ti / .tj / ¿ C 2 C \ D for i j , then xk 1 0. ¤ C D Using this lemma we prove the following auxiliary statement.

Lemma 2.5. Let F be a field of zero characteristic such that the transcendence degree of F over Q is finite. If the automorphism ı of the field F acts on the elements z1, z2;::: of the field F by the rule

ı zi ˛ai zi ; W 7! where ˛ F , 1 ai Q F and .ai / .aj / ¿ for i j , then there are 2 ¤ 2 Â \ D ¤ only a ﬁnite number of non-zero elements in the set z1; z2;::: .

Proof. If all the elements z1; z2;::: are equal to zero, there is nothing to prove. Hence we can assume that there exists a non-zero element in the set z1; z2;::: . Without loss of generality we can assume that z1 0. (Otherwise we can reorder ¤ the elements z1; z2;::: such that the ﬁrst element is not equal to zero. If the statement holds for the reordered set, then it holds for the original set z1; z2;:::). 1 Denote by yi zi z . Then the automorphism ı acts on the element yi by the D 1 rule 1 1 ı.yi / ı.zi z / ı.zi /ı.z / D 1 D 1 1 1 1 ˛ai zi ˛ a z D 1 1 1 1 1 ai a zi z ai a yi : D 1 1 D 1 906 A. Fel’shtyn and T. Nasybullov

Since the transcendence degree of F over Q is finite, there exists a maximal subset of algebraically independent over Q elements in the set y2; y3;::: , i.e. there exists a finite set yi1 ; yi2 ; : : : ; yik of elements algebraically independent over Q such that the set yi1 ; yi2 ; : : : ; yik ; yj is algebraically dependent over Q for every j . Without loss of generality we can assume that the set y2; : : : ; yk is a maximal subset of elements algebraically independent over Q in the set y1; y2;::: . If n > k is a positive integer, then the elements y2; : : : ; yk; yn F satisfy the 2 1 conditions of Lemma 2.4. Thus yn 0 for all n > k and since yn znz , we D D 1 have zn 0 for all n > k and the only non-zero elements are z1; z2; : : : ; z . D k Let us recall some facts about Chevalley groups. We use definitions and nota- tions from [4]. Let ˆ be an indecomposable root system of rank l with a subsystem of simple roots , l. The elementary Chevalley group ˆ.F / of type ˆ over a field F j j D is a subgroup in the automorphism group of a simple Lie algebra L of type ˆ, which is generated by the elementary root elements x˛.t/, ˛ ˆ, t F . The 2 2 dimension of the Lie algebra L is equal to ˆ and therefore the group ˆ.F / j j C j j can be considered as a subgroup in the group of all . ˆ / . ˆ / j j C j j j j C j j invertible matrices. In the elementary Chevalley group, we consider the following important elements: 1 n˛.t/ x˛.t/x ˛. t /x˛.t/; h˛.t/ n˛.t/n˛. 1/; t F ; ˛ ˆ: D D 2 2 For an arbitrary Chevalley group G of type ˆ over a field F we have the following short exact sequence of groups: 1 Z.G/ G ˆ.F / 1; ! ! ! ! where Z.G/ is the center of the group G, and by Lemma 2.2 we are mostly inter- ested in the study of the R -property for elementary Chevalley groups. Detailed information about1 automorphisms of Chevalley groups can be found in [30, 43]. Every Chevalley group has the following automorphisms:

(1) Inner automorphism 'g , induced by an element g G, 2 1 'g x gxg : W 7! (2) Diagonal automorphism 'h, ' x hxh 1; h W 7! where the element h can be presented as a diagonal . ˆ / . ˆ / j j C j j j j C j j matrix. If F is an algebraically closed ﬁeld, then any diagonal automorphism is inner [43, Lemma 4]. The R and S properties for linear algebraic groups 907 1 1

(3) Field automorphism ı,

ı x .xij / .ı.xij //; W D 7! where ı is an automorphism of the field F . (4) Graph automorphism , which acts on the generators of the group G by the rule x˛.t/ x .t/; W 7! .˛/ where is a symmetry of a Dynkin diagram. The order of the graph automorphism is equal to 2 or to 3. Any field automorphism commutes with any graph automorphism. All diagonal automorphisms form a normal subgroup in the group which is generated by diagonal, graph and field automorphisms. A theorem of Steinberg says that for every automorphism ' of the elementary Chevalley group G ˆ.F / there exists an inner automorphism 'g , a diagonal D automorphism 'h, a graph automorphism and a field automorphism ı such that ' ı' 'g (see [30]). D h 3 Chevalley groups

In this section we extend the following result from [43, Theorem 1].

Theorem 3.1. Let G be a Chevalley group of type ˆ over a ﬁeld F of zero characteristic and the transcendence degree of F over Q is ﬁnite. Then:

(1) If ˆ has one of the types A .l 7/, B .l 4/, E8, F4, G2, then G possesses l l the R -property. 1 (2) If the equation T k a can be solved in the ﬁeld F for every element a, then D G possesses the R -property also for the root systems Al .l 2; 3; 4; 5; 6/, 1 D Bl .l 2; 3/, Cl .l 3/, Dl .l 4/, E6, E7, where k is a positive integer from theD following table.

ˆ Al Bl Cl Dl E6 E7 k l 1 2 2 2 6 2 C In particular, this theorem says that if F is an algebraically closed field of zero characteristic such that the transcendence degree of F over Q is finite, then a Chevalley group of any normal type over the field F possesses the R -property. Here we exclude the condition of solvability of equations from the second1 item of Theorem 3.1. We prove the following result. 908 A. Fel’shtyn and T. Nasybullov

Theorem 3.2. Let G be a Chevalley group of type ˆ over a field F of zero characteristic. If the transcendence degree of F over Q is finite, then G possesses the R -property. 1 Proof. Since G=Z.G/ ˆ.F /, by Lemma 2.2 it is sufficient to prove that the ele- Š mentary Chevalley group ˆ.F / possesses the R -property. It suffices to assume that G ˆ.F /. 1 D Let us consider an arbitrary automorphism ' of the group G and prove that the number of '-conjugacy classes is infinite. By the theorem of Steinberg there exists an inner automorphism 'g , a diagonal automorphism 'h, a graph automorphism and a field automorphism ı such that

' ı' 'g : D h By Lemma 2.1 the Reidemeister number R.'/ is inﬁnite if and only if the Reide- meister number R.''g 1 / is inﬁnite, and we can consider that ' ı'h. D Suppose that R.'/ < and consider the following elements of the group G: 1 gi h˛ .pi1/h˛ .pi2/ : : : h˛ .p /; i 1; 2; : : : ; D 1 2 l il D where p11 < p12 < < p < p21 < p22 < are prime numbers. In matrix 1l representation the element gi has diagonal form

gi diag.ai1; ai2; : : : ; ai ˆ ; 1; : : : ; 1/; D j j „ ƒ‚ … ˆ j j for certain rational numbers aij such that .aij / ¿ and .aij / .ars/ ¿ ¤ \ D for i r since .aij / pi1; : : : ; p (see [43]). ¤ Â ¹ il º Since R.'/ < , there exists an inﬁnite subset of '-conjugated elements in 1 the set g1; g2;::: . Without loss of generality we can assume that all the elements g1; g2;::: belong to the '-conjugacy class Œg1' of the element g1. Then there exists an inﬁnite set of matrices Z2;Z3;::: from G such that 1 g1 Zi gi '.Z /; i 2; 3; : : : . D i D Acting on these equalities by iterations of the automorphism ' we have

1 g1 Zi gi '.Z /; D i 2 1 '.g1/ '.Zi /'.gi /' .Z /; D i 2 2 2 3 1 ' .g1/ ' .Zi /' .gi /' .Zi /; D: : 5 5 5 6 1 ' .g1/ ' .Zi /' .gi /' .Z /: i 2; 3; : : : . D i D The R and S properties for linear algebraic groups 909 1 1

If we multiply all of these equalities, we conclude that 5 5 6 1 g1'.g1/:::' .g1/ Zi gi '.gi /:::' .gi /' .Z /: (3.1) D i Since the matrix gi has a diagonal form and the automorphism 'h acts as con- jugation by the diagonal matrix, we have ' .gi / gi . Since the matrix gi has h D rational entries, it follows that ı.gi / gi and therefore '.gi / .gi /. If we 5 D 5 D denote gi gi '.gi /:::' .gi / gi .gi /::: .gi /, then Q D D gi diag.bi1; bi2; : : : ; bi ˆ ; 1; : : : ; 1/; i 1; 2; : : : ; Q D j j „ ƒ‚ … D l since permutes diagonal elements of the matrix gi . Moreover, .bij / ¿ and ¤ .bij / .brs/ ¿ for i r, since .bij / .ai1/ .ai ˆ /. \ D ¤ Â [ [ j j Since graph and ﬁeld automorphisms commute and diagonal automorphisms form a normal subgroup in the group, which is generated by graph, ﬁeld and diagonal automorphisms, for a certain diagonal automorphism 'h we have Q 6 6 6 6 ' .ı'h/ 'hı : D D Q Since the order of the automorphism is equal to 2 or to 3, it follows that 6 id 6 6 D and ' 'hı . Then equality (3.1) can be rewritten D Q 6 1 6 1 6 1 1 g1 Zi gi ' .Zi / Zi gi 'hı .Zi / Zi gi hı .Zi /h ; i 2; 3; : : : . Q D Q D Q Q D Q Q Q D If we multiply this equality by the element h on the right and denote gi gi h, Q O DQ Q then we have 6 1 g1 Zi gi ı .Z /; i 2; 3; : : : . (3.2) O D O i D From this equality we have

6 1 ı .Zi / g Zi gi ; i 2; 3; : : : . (3.3) DO1 O D If we denote h diag.c1; c2; : : : ; c ˆ ; 1; : : : ; 1/; Q D j j „ ƒ‚ … l then

gi gi h diag.bi1c1; bi2c2; : : : ; bi ˆ c ˆ ; 1; : : : ; 1/; i 2; 3; : : : . O DQ Q D j j j j „ ƒ‚ … D l Let ! Qi Ri Zi ; D Si Ti where Qi .qi;mn/ is a ˆ ˆ matrix, Ri .ri;mn/ is a ˆ matrix, D j j j j D j j j j Si .si;mn/ is a ˆ matrix, Ti .ti;mn/ is a matrix. Then by D j j j j D j j j j 910 A. Fel’shtyn and T. Nasybullov equality (3.3) for all m 1; : : : ; ˆ , n 1; : : : ; ˆ we have D j j D j j 6 1 ı .qi;mn/ .b1mcm/ bincnqi;mn dmnbinqi;mn; i 2; 3; : : : ; D D D 1 where dmn .b1mcm/ cn. Since .bin/ ¿ and .bin/ .bj n/ ¿ for all D ¤ \ D i; j with i j , we can apply Lemma 2.5 to the set q2;mn; q3;mn;::: . There- ¤ fore by Lemma 2.5 there exists a positive integer Nmn such that qi;mn 0 for D every i > Nmn. If we denote by N the value

N max Nmn; D n; m 1;:::; ˆ D j j then for every i > n we have Qi O ˆ ˆ . Using the same arguments for the D j jj j matrices S2;S3;::: , we conclude that for sufﬁciently large indexes i all the matrices Si are matrices with zero entries only, and therefore the matrix Zi has the form ! O ˆ ˆ Ri Zi j jj j : D O ˆ Ti j jj j The determinant of this matrix is equal to zero, therefore Zi cannot belong to G. This contradiction proves the theorem.

4 Linear algebraic groups

If G is a linear algebraic group over an algebraically closed ﬁeld, then it has a unique maximal solvable normal subgroup R.G/, called the radical of G. A connected linear algebraic group G is called reductive if its radical is a torus, or, equivalently, if it can be decomposed G G T with G a semisimple group and D 0 0 0 T 0 a central torus [52, Section 6.5]. The quotient group G=R.G/ has a trivial radical, i.e. is a semisimple group [31, Section 19.5].

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/, then G possesses the R -property. 1 Proof. For the group G we have the following short exact sequence of groups: 1 R.G/ G G=R.G/ 1: ! ! ! ! Since G is reductive, the radical R.G/ is a central torus and therefore is a characteristic subgroup of G. Hence by Lemma 2.2 it is sufficient to prove that the The R and S properties for linear algebraic groups 911 1 1 semisimple group G=R.G/ possesses the R -property and we can assume that G is a semisimple linear algebraic group. Every1 semisimple linear algebraic group is a product, with some amalgamation of (finite) centers, of its simple subgroups H1;H2;:::;Hk (see [31, Section 14.2])

G H1 H : D ı ı k Every simple linear algebraic group Hi is a Chevalley group of (normal) type ˆi over the ﬁeld F . Factoring the group G by its center we have the following short exact sequence of groups:

1 Z.H1 H / H1 H ˆ1.F / ˆ .F / 1; ! ı ı k ! ı ı k ! k ! where î .F / is the elementary Chevalley group of type î over the field F . Hence, by Lemma 2.2 we can assume that G ˆ1.F / ˆ .F / and prove D k that this group possesses the R -property. Permute the groups ˆ1.F /; : : : ; ˆk.F / so that all groups with the same1 root system form blocks

G ˆ1.F / ˆ1.F / ˆ2.F / ˆ2.F / ˆr .F / ˆr .F /; D „ ƒ‚ … „ ƒ‚ … „ ƒ‚ … k1 k2 kr where k1 k2 kr k. Denote by C C C D Gi ˆi .F / ˆi .F /: D „ ƒ‚ … ki

Every group Gi is a characteristic subgroup of G G1 Gr . Therefore D by Lemma 2.2 it is sufﬁcient to prove that some group Gi possesses the R -property. Thus we can consider that G ˆ.F / ˆ.F / ˆ.F /k. 1 D D Every element g G ˆ.F /k can be presented as a direct sum of k matrices 2 D g1; : : : ; g of the size . ˆ / . ˆ / each of which belongs to ˆ.F /. k j j C j j j j C j j The automorphism group of G has the form Aut.G/ .Aut.ˆ.F ///k h S ; (4.1) D k where Sk is the full permutation group on k symbols. To prove that the group G ˆ.F /k possesses the R -property, consider an D 1 arbitrary automorphism ' of the group G and we will prove that R.'/ . By D 1 equality (4.1) the automorphism ' can be written in the form

' .'1;:::;' ; /; D k where '1;:::;' Aut.ˆ.F //, S , and ' acts on the group G by the rule k 2 2 k ' x1 x2 x '1 .x1 / '2 .x2 / ' .x /; (4.2) W ˚ ˚ ˚ k 7! ˚ ˚ ˚ k k where i denotes the image of i by the permutation . 912 A. Fel’shtyn and T. Nasybullov

Every automorphism 'i Aut.ˆ.F // can be presented as a product of an inner 2 automorphism 'gi , a diagonal automorphism 'hi , a graph automorphism i and a ﬁeld automorphism ıi . Since F is an algebraically closed ﬁeld, every diagonal automorphism 'hi is inner [43, Lemma 4], hence for every i we can assume that 'i 'x ıi . Then the automorphism ' can be presented as a product of two D i i automorphisms

' .'x ;'x ;:::;'x ; id/. ı1; ı2;:::; ı ; /; D 1 2 k 1 2 k k where .'x1 ;'x2 ;:::;'xk ; id/ is an inner automorphism. By Lemma 2.1 we can consider that 'i ıi and D i ' . ı1; ı2;:::; ı ; /: D 1 2 k k Using induction on r, we will prove that r ' g1 g 1.x r / .x r /; (4.3) W ˚ ˚ k 7! 1 ˚ ˚ k k where i 'i ' 2 :::' r . D i i The basis of induction (r 1) is obvious (equality (4.2)). If we suppose that D equality (4.3) holds for some r, then r 1 r ' .g1 g / '.' .g1 g // C ˚ ˚ k D ˚ ˚ k '. 1.x r / .x r // D 1 ˚ ˚ k k '1 1 .x r 1 / 'k k .x r 1 /: D 1 C ˚ ˚ k C

Noting that 'i i 'i ' 2 :::' r 1 we obtain equality (4.3). D i i C Consider the set of elements g1; g2;::: of the group ˆ.F / from Theorem 3.2

gi h˛ .pi1/h˛ .pi2/ : : : h˛ .p /; i 1; 2; : : : ; D 1 2 l il D where p11 < p12 < < p < p21 < p22 < are prime integers. These ele- 1l ments are presented by diagonal matrices

gi diag.ai1; ai2; : : : ; ai ˆ ; 1; : : : ; 1/; i 1; 2; : : : ; D j j „ ƒ‚ … D l where aij are rational numbers such that .aij / ¿ and .aij / .ars/ ¿ ¤ \ D for i r. ¤ As already shown in Theorem 3.2, for every automorphism 'j ıj we have D j 'j .gi / j .gi /. D k Let us consider the set of elements g1; g2;::: of the group G ˆ.F / , where Q Q D gi gi gi . Then by the arguments above Q D ˚ ˚ '.gi / .gi / .gi /: Q D 1 ˚ ˚ k The R and S properties for linear algebraic groups 913 1 1

Suppose that R.'/ < . Then there is an inﬁnite subset of '-conjugated ele- 1 ments in the set g1; g2;::: . Without loss of generality we can consider that all Q Q matrices g1; g2;::: belong to the '-conjugacy class Œg1' of the element g1. Then Q Q Q Q for certain matrices Z2;Z3;::: we have 1 g1 Zi gi '.Z /; i 2; 3; : : : . Q D Q i D Denote by s the order of the permutation and act on this equality by iterates of the automorphism ' 1 g1 Zi gi '.Z /; Q D Q i 2 1 '.g1/ '.Zi /'.gi /' .Zi /; Q D: Q : 6s 2 6s 2 6s 2 6s 1 1 ' .g1/ ' .Zi /' .gi /' .Z /; Q D Q i 6s 1 6s 1 6s 1 6s 1 ' .g1/ ' .Zi /' .gi /' .Z /: Q D Q i If we multiply all of these equalities, we obtain the following equality: 2 6s 1 g1'.g1/' .g1/:::' .g1/ Q Q Q Q (4.4) 2 6s 1 6s 1 Zi gi '.gi /' .gi /:::' .gi /' .Z /: D Q Q Q Q i 2 6s 1 The element gi '.gi /' .gi /:::' .gi / can be rewritten in details as Q Q Q Q 2 6s 1 gi '.gi /' .gi /:::' .gi / .gi gi /. .gi / .gi // Q Q Q Q D ˚ ˚ 1 ˚ ˚ k :::. 6s 1 .gi / 6s 1 .gi // 1 ˚ ˚ k gi 1 .gi /::: 6s 1 .gi / gi k .gi / D 1 ˚ ˚

::: 6s 1 .gi / k

gi1 g ; D b ˚ ˚ bik where gij gi j .gi /::: 6s 1 .gi /. b D j Since every graph automorphism j permutes elements on the diagonal of the matrix gi , we have for every j 1; : : : ; k and i 1; 2; : : : that D D bgij diag.bij1; bij 2; : : : ; bij ˆ ; 1; : : : ; 1/; (4.5) D j j „ ƒ‚ … l where .bijr / ¿ and .bijr / .buvw / ¿ for all i; u with i u since .bijr / ¤ \ D ¤ is a subset of pi1; : : : ; pil . ¹ s º From (4.3) we have ' . 1; 2; : : : ; ; id/, where i 'i ' 2 :::' r . D k D i i Since all automorphisms '1;'2;:::;'k are products of graph and ﬁeld automorphisms ('i ıi ) and graph and ﬁeld automorphisms commute, it follows that D i 914 A. Fel’shtyn and T. Nasybullov

every automorphism i is a product of graph and field automorphisms i i D i for certain i , i . Therefore 6s s 6 6 6 6 6 6 6 6 ' .' / . 1;::: ; id/ . ;::: ; id/ . ;::: ; id/: D D 1 k k D 1 1 k k D 1 k Using this fact, denoting by Zi Zi1 Z projecting equality (4.4) to the D ˚ ˚ ik first group ˆ.F / we obtain the equality 6 g11 Zi1gi1 .Zi1/; i 2; 3; : : : . b D b 1 D This equality is the same as equality (3.2) from Theorem 3.2. Using the same arguments as in Theorem 3.2, we conclude that for sufficiently large number N the matrix ZN is degenerate but this contradicts the fact that ZN belongs to G. We use the fact that the group G is a reductive linear algebraic group in order to say that the radical R.G/ is a characteristic subgroup of G. Even Theorem 4.1 holds for every connected linear algebraic group such that the radical R.G/ is a characteristic subgroup. For example, if every automorphism of the group G is a morphism of the group G (as in the case of an affine manifold), then the radical R.G/ is characteristic [52, Theorem 7.1 (c)] and Theorem 4.1 holds for such groups.

5 Finite Reidemeister number in linear groups

Following [47], we define the Reidemeister spectrum of G as Spec.G/ R.'/ ' Aut.G/ : D ¹ W 2 º In particular, G possesses the R -property if and only if Spec.G/ . 1 D ¹1º It is easy to see that Spec.Z/ 2 , and, for n 2, the spectrum of Zn D ¹ º [ ¹1º is full, i.e. Spec.Zn/ N . For free nilpotent groups we have the following: D [ ¹1º Spec.N22/ 2N (N22 is the discrete Heisenberg group) ([14, 32, 47]), D [ ¹1º 2 Spec.N23/ 2k k N ([47]), D ¹ W 2 º [ ¹1º Spec.N32/ 2k 1 k N 4k k N ([47]). D ¹ W 2 º [ ¹ W 2 º [ ¹1º Recently, in [5, 47] it was proven that the group Nrc .r > 1/ admits an automorphism with finite Reidemeister number if and only if c < 2r. In [24], examples of polycyclic non-virtually nilpotent groups which admit automorphisms with finite Reidemeister numbers have been described. In these examples G is the semidirect product of Z2 and Z by the Anosov automorphism 2 1 defined by the matrix . 1 1 /. The group G is solvable and of exponential growth. 0 1 2 The automorphism ' with finite Reidemeister number is defined by . 1 0 / on Z and as id on Z. The R and S properties for linear algebraic groups 915 1 1

Metabelian (therefore, solvable) finitely generated, non-polycyclic groups have quite interesting Reidemeister spectrum [12]: for example, if the homomorphism Z Aut.ZŒ 1 2/ is such that .1/ . r 0 /, then we have the following cases: W ! p D 0 s (a) If r s 1, then D D ˙ Spec.ZŒ1=p2 Ì Z/ 2n n N; .n; p/ 1 ; D ¹ W 2 D º [ ¹1º where .n; p/ denotes the greatest common divisor of n and p. (b) If r s 1, then D D ˙ Spec.ZŒ1=p2 Ì Z/ 2pl .pk 1/; 4pl l; k > 0 : D ¹ ˙ W º [ ¹1º (c) If rs 1 and r 1, then D j j ¤ Spec.ZŒ1=p2 Ì Z/ 2.pl 1/; 4 l > 0 : D ¹ ˙ W º [ ¹1º (d) If either r or s does not equal to 1, and rs 1, then ˙ ¤ Spec.ZŒ1=p2 Ì Z/ : D ¹1º In [33] Jabara proved that if a residually finite group G admits an automorphism of prime order p with finite Reidemeister number, then G is a virtually nilpotent group of class bounded by a function of p. On the other hand, we have described in the Introduction (Section 1) a lot of classes of non-virtually-solvable, finitely generated, residually finite groups which have the R -property. Taking these ideas together was a motivation for the following conjecture.1

Conjecture 5.1 ([18, Conjecture R]). Every residually finite, finitely generated group either possesses the R -property or is a virtually solvable group. 1 Here we study this question for infinite linear groups.

Proposition 5.2. Let G be a reductive linear algebraic group over a field F of zero characteristic and finite transcendence degree over Q. If G possesses an automorphism ' with finite Reidemeister number, then G is a torus.

Proof. Since G possesses an automorphism ' with ﬁnite Reidemeister number, by Theorem 4.1, the quotient group G=R.G/ is trivial, therefore G R.G/ and D hence G is a central torus (therefore, is solvable). 916 A. Fel’shtyn and T. Nasybullov

6 Groups with property S 1 Let ‰ Out.G/ Aut.G/=Inn.G/, and let S.‰/ be the set of isogredience 2 D classes of ‰. Then S.Id/ can be identified with the set of conjugacy classes of G=Z.G/ (see [18]). The definition of similarity (isogredience) from Section 1 goes back to Jacob Nielsen. He observed (see [34]) that conjugate lifting of homeomorphisms of sur- faces have similar dynamical properties. This led Nielsen to the definition of the isogredience of liftings in this case. Later Reidemeister and Wecken succeeded in generalizing the theory to continuous maps of compact polyhedra (see [34]). The set of isogredience classes of automorphisms representing a given outer automorphism and the notion of index Ind.‰/ defined via the set of isogredience classes are strongly related to important structural properties of ‰ (see [19]). One of the main results of [38] is that for any non-elementary hyperbolic group and any ‰ the set S.‰/ is infinite, i.e. S.‰/ . Thus, this result says: any D 1 non-elementary hyperbolic group possesses the S -property. On the other hand, finitely generated abelian groups are evidently non-1S -groups. 1 Two representatives of ‰ have forms 'sa, 'qa for some s; q G and fixed 2 a ‰. They are isogredient if and only if 2 1 'qa 'g 'sa'g 'g 's'a.g 1/a; D D 1 'q 'gsa.g 1/; q gsa.g /c; c Z.G/ D D 2 (see [38, p. 512]). So, the following statement is proved.

Lemma 6.1 ([18, Lemma 3.3]). Let ' ‰ be an automorphism of the group G and 2 let ' be an automorphism of the group G=Z.G/ which is induced by '. Then the number S.‰/ is equal to the number of '-conjugacy classes in the group G=Z.G/. Since Z.G/ is a characteristic subgroup, we obtain the following statement.

Theorem 6.2 ([18, Theorem 3.4]). Let G be a group with finite center. Then G possesses the R -property if and only if G possesses the S -property. 1 1 A more advanced example of a non-S -group is Osin’s group [46]. This is a non-residually finite exponential growth group1 with two conjugacy classes. Since it is simple, it cannot possess the S -property (see [18]). 1 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 algebraic group G over the field F has a nontrivial quotient group G=R.G/, then G possesses the S -property. 1 The R and S properties for linear algebraic groups 917 1 1

Proof. Since R.G/ is a characteristic subgroup of G, by Lemma 2.3 it is sufﬁcient to prove the theorem for semisimple group G=R.G/. The result follows immedi- ately from Theorem 4.1 and Theorem 6.2 and from the fact that a semisimple linear algebraic group has ﬁnite center.

Proposition 6.4. Let G be a reductive linear algebraic group over a field F of zero characteristic and finite transcendence degree over Q. If G possesses an outer automorphism ‰ with finite number S.‰/, then G is a torus.

Proof. Since G possesses an outer automorphism ‰ with ﬁnite number S.‰/, it follows from Theorem 6.3 that it has a trivial quotient G=R.G/, thus G R.G/ D and is a central torus.

The conjecture of Fel’shtyn and Troitsky from Section 5 can be rewritten in terms of the S -property by the following way. 1 Conjecture 6.5 ([18, Conjecture S]). Every residually finite, finitely generated group either possesses the S -property or is a virtually solvable group. 1 Really, if S.'Inn.G// < for some automorphism ' Aut.G/, then by Lem- 1 2 ma 6.1 we have R.'/ < , where ' is an automorphism of the group G=Z.G/ 1 induced by '. Since G is a residually finite finitely generated group, G=Z.G/ is also finitely generated and residually finite and by Conjecture 5.1 is a virtually solvable group. This means that there exists a solvable subgroup H G=Z.G/ of finite index. Ä Let n be a derived length of H, i.e. H .n/ 1. Let H be the full preimage of D H under the canonical homomorphism G G=Z.G/. Then H .n/ Z.G/ and ! Ä H .n 1/ 1, therefore H is a solvable group. Since C D G=H .G=Z.G// = .H=Z.G// .G=Z.G// = H; ' D the index of H in G is equal to the index of H in G=Z.G/, i.e. is finite, therefore G is a virtually solvable group. In all, we have proven that [18, Conjecture S] can be formulated without the restriction that the group under consideration has finite center.

Acknowledgments. The authors are grateful to Andrzej Da¸browski, Evgenij Troitsky and Evgeny Vdovin for the numerous important discussions on linear algebraic groups. The ﬁrst author would like to thank the Max Planck Institute for Mathematics (Bonn) for its kind support and hospitality while a part of this work was completed. 918 A. Fel’shtyn and T. Nasybullov

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Received June 17, 2015.

Author information Alexander Fel’shtyn, Instytut Matematyki, Uniwersytet Szczecinski, ul. Wielkopolska 15, 70-451, Szczecin, Poland. E-mail: [email protected] Timur Nasybullov, Sobolev Institute of Mathematics, ak. Koptyug avenue 4, 630090, Novosibirsk, Russia. E-mail: [email protected]