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МАТЕМАТИКА
Вестн. Ом. ун-та. 2016. № 1. С. 22-25.
УДК 512.5 V.A. Roman'kov
A SHORT SURVEY ON LINEARITY OF AUTOMORPHISM GROUPS OF ALGEBRAS AND GROUPS
The main purpose of this paper is to describe some known results and outline corresponding approaches which when applied to automorphism groups of algebras or groups establishes that these groups are linear or non-linear.
Keywords: automorphism group; linear representation; free group; solvable group; relatively free group; relatively free algebra.
1. On linearity of automorphism groups of groups.
There are some useful tests for linearity, such as
(1) A linear group has ascending chain condition on centralizers.
(2) (Mal'cev) Finitely generated (f.g.) linear groups are residually finite.
(3) (Mal'cev) A solvable linear group is nilpotent-by-(abelian-by-finite).
(4) (Tits) A linear group either contains a free group of rank two, or is solv-able-by-(locally finite).
1.1. Non-linear automorphism groups of f.g. solvable groups.
In [1], Tits showed that a finitely generated matrix group either contains a solvable normal subgroup of finite index (i.e., is almost solvable) or else contains a noncyclic free subgroup.
In [2],Bachmuth and Mochizuki conjectured that Tits' alternative is satisfied in any f.g. group of automorphisms of af.g. solvable group. They point out that their conjecture holds for solvableabelian-by-nilpotent groups and in some other cases. It turned out that this conjecture breaks down, in general. It has been independently shown by Hartley [3] and the author [4] with absolutely different approaches.
We will give these results and corresponding approaches starting with [4].
Theorem 1 (Roman'kov [4]. The direct wreath product of the group of IA-automorphisms of an arbitrary f.g. solvable group with an infinite cyclic group is embeddable in the group of automorphisms of some f.g. solvable group.
We recall that the IA-automorphisms of a group G are those automorphisms which are identical modulo the commutator group G'.
Corollary 1. If A is an arbitrary f.g. almost solvable group, then the wreath product B =AwrZ is embeddable in the group of automorphismsof some f.g. solvable group.
The deduced corollary yields a negative answer to Bachmuth and Mochi-zuki's question since, under its conditions, Tits' alternative is satisfied in the group G if and only if A is solvable.
In [2], Bachmuth and Mochizuki also conjectured that in a f.g. group of automorphisms of a f.g. solvable group one can find a subnormal series of finite length whose factors are either abelian or are matrix groups over commutative Noetherian rings.
Corollary 2. There exists af.g. group of automorphisms of a f.g. solvable group which does not have the indicated property.
The infinite countable direct power Gzof a simple finite non-abelian group G is not embeddable in a group with the indicated subnormal series. It is easy to prove this by induction on the length of the series using the remark of Merzlyakov that the direct product of an infinite number of non-abelian groups cannot be represented by matrices (see [5, Example 3]). The proofs for fields and for commutative Noetherian rings are identical. This power Gz is the basic
© V.A. Roman'kov, 2016
subgroup of G wr Z. By Corollary 2, the latter group is embeddable in the group of automorphisms of some f.g. solvable group.
A group is called perfect if it equals its derived subgroup. In [3], a group G is called perfectly distributed, if every subgroup of finite index of G contains a non-trivial f.g. perfect subgroup. Clearly no perfectly distributed group can be soluble-by-finite.
Theorem 2 (Hartley [3]). There exists a f.g. solvable group G of derived length three whose automorphism group contains subgroups K < L such that L is f.g., L/K is infinite cyclic, K is perfectly distributed and locally finite.
Moreover, it can be even arranged that K is locally a direct power of any finite non-abelian simple group. Clearly L is not solvable-by-finite, nor does it contain a non-abelian free subgroup.
The group G of Theorem 2 is actually an extension of a locally finite group by an infinite cyclic group. But a somewhat more complicated version of the construction for Theorem 2 gives
Theorem 3 (Hartley [3]). There exists a f.g. group solvable group G of derived length four whose automorphism group contains a torsionfree subgroup L having a normal subgroup K such thatL is f.g., L/K is infinite cyclic, K is perfectly distributed, every f.g. subgroup of K is abelian-by-finite.
1.2. On linearity of automorphism groups of f.g. free groups
Let Fn be an absolutely free group with basis Xn = {x1,...,xn}.
Theorem 4.
(a) (Krammer [6]) AutF2 is linear.
(b) (Formanek and Procesi [7]).AutiJ1 is not linear for n > 3.
The proof of (b) in [7] uses the representation theory of algebraic groups to show that a kind of diophantine equation between the irreducible representations of a group G is impossible unless G is abelian-by-finite. This leads to statement, which says that the HNN-extension H(G) = < G x G, t| t(g,g)t"! =
= (1,g) for all g in G> (1)
cannot be a linear group if G is not nilpotent-by-abelian-by-finite. The statement (b) then is proved by showing that for n > 3, the automorphism group of a free group of rank n contains H(F2).
Refer as a unitriangularautomorphism of Fn with respect to Xn to every automorphism^ determined by a mapping of the form
9 xu xt^ vtxt for i = 2, ..., n, (2) where^j = vt (%,...,J is an arbitrary element of Fi_1. Every collection (v2,...,v„)6
F1x—x Fn_i determines an automorphism 9of Fn. The unitriangularautomorphisms of Fn constitute a subgroup UTAut/^, which we call the group of unitriangularautomorphisms of Fn. To simplify expressions, we denote it by Vn. It is easy to see that, up to isomorphism, Vn is independent of the choice of Xn.
Theorem 5 (Romankov [8]). The group l^of unitriangular automorphisms of the free group Fn of rank n is linear if and only if n < 3.
As V1 is trivial and V2 is infinite cyclic, these groups are linear. In [8], it is shown that V3is isomorphic to a subgroup of AutF2. Then by Krammer's theorem V3 is linear.
We need in two statements about any group G that follow.
Theorem 6.
(a) (Formanek, Procesi [7]).Let p be a linear representation of H(G). Then the image of G x {1} has a subgroup of finite index with nilpotent derived subgroup, i.e., is nilpotent-by-abelian-by-finite.
(b) (Brendle, Hamidi-Tehrani [9]). Let N be a normal subgroup of H(G) such that the image of
G x {1} in H(G)/N is not nilpotent-by-abe-lian-by-finite. Then H(G)/N is not linear.
In [8], a subgroup G of V4 has been constructed that is isomorphic to a quotient H(F2)/N such that the image of G x {1} in H(F2)/N is not nilpotent-by-abelian-by-finite.
1.3. On linearity of automorphism groups of f.g. relatively free groups.
Given an arbitrary variety C of groups, denote by Gn the free group in C with a fixed basis Yn = {yi,...,yn}. Refer as a unitriangular automorphism of Gnwith respect to Ynto every automorphism 0 determined by a mapping of the form 0 : yi^ yi, Vi^iVi for i = 2, ..., n, (3) where ut = ut (yi,...,y;-i) is an arbitrary element of Gi_i. Every collection (u2,...,un) £G1x---xGn ldetermines an automorphism0 of Gn. The unitriangular automorphisms of Gn constitute a subgroup UTAutGn, which we call the group of unitriangular automorphisms of Gn. To simplify expressions, we denote it by Un. It is easy to see that, up to isomorphism, Un is inde-pendentof the choice of Yn.
Theorem 7 (Auslander and Baumslag[10]). The automorphism group of every finitely generated nilpotent-by-finite group is linear.
Moreover, the holomorph of a group of this type (hence, its automorphism group as well) admits a faithful matrix representation over the ring Z of integers. This implies in particular that the automorphism groups of finite rank relatively free groups in nilpotent-by-finite varieties of groups admit faithful matrix representations.
Theorem 8 (Olshanskii [11]).A relatively free group Gn is neither free nor nilpotent-by-finite then AutGn is not linear.
Observe also that AutF^ for every nontrivial variety C includes a subgroup consisting of all permutations of an infinite set of free generators, which is isomorphic to the symmetric group on the infinite set. It is well known that the latter is not linear. Olshanskii's proof of Theorem 8 uses the following reasoning.
For every group G there is a homomor-phism o : G ^ Aut G associating to each h the inner automorphismo(h): f ^ hfh"1. The kernel of o is the center C(G) of G and the image of o
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V.A. Roman'kov
is the group Inn G of inner automorphisms of G. The latter is normal in Aut G, which means that we may regard the quotient G/C(G) ~ Inn G as a normal subgroup of Aut G. It is shown in [11] that in the case of neither free nor nilpo-tent-by-finite relatively free group Gn there exists an automorphism0 of Gn such that the extension P of Gn/C(Gn) by means of 0 is not linear. Furthermore, 0 was chosen so that the eigenvalues of the induced linear transformation (Gn)ab = Gn/Gn' of abelianization, which in this case, on assuming that Aut Gn is a linear group, is a rank n free abelian group, were not roots of unity. The automorphism0is certainly not uni-triangular since all its eigenvalues are equal to 1. The group Inn Gn/C(Gn) also fails to consist of unitriangular automorphisms. Thus, the result of Olshanskiiand his method of proof give no information on the possiblelinearity of the group Un= UTAutGn.
Theorem 9 (Erofeev and Roman'kov [12]).
(a) Let Gn be a rank n > 2 relatively free group in an arbitrary variety C of groups. The following hold:U1 is trivial, while U2is a cyclic group of the order equal to the exponent of C . These groups admit faithful matrix representations. For n > 3, if Gn_! is a nilpotent group then so is Un. If Gn_! is nilpotent-by-finite then Un is linear over Z.
(b) Let Gn be a rank n > 3 relatively free group in an arbitrary nontrivial variety C of groups different from the variety of all groups. If Gn_! is not a nilpotent-by-finite group then the group Un = UTAutGnof unitriangular automorphisms of Gnadmits no faithful matrix representation over any field.
Thus, the claims of Theorem 9 yield exhaustive information on the linearity of groups of unitriangularautomorphisms of relatively free groups of finite rank in the proper varieties of groups. Namely, Un admits a faithful matrix representation over some field if and only if Gn^is a nilpotent-by-finite group.
2. On linearity of automorphism groups of relatively free algebras.
Let K be a field. We restrict exposition to considering the following classical algebras over K (the subscript n > 2 stands for the rank, i.e., the power of a set of free generators): the free Lie algebra Ln, the free associative algebra An, the absolutely free algebra Fn, and the algebra Pn of polynomials. It is irrelevant whether we consider these algebras with or without identity.
The automorphism group and the tame automorphism group of an algebra Cn are denoted by
AutCn and TAutCn, respectively. By definition, the subgroup TAutCnis generated in AutCnby all elementary automorphisms and all nondegenerate linear changes of generators. Respectively, TAutCnis called the subgroup of
tame automorphisms. By definition, the elementary automorphismsare of the form
4> : x; ^ a x; + f(x1,...,xi_1.xi+1,...,xn),xj ^
^ Xj for i* j, (4)
where i,j = 1,...,n, a is a nonzero element of the field K, and f(x1,...,x;_1,,..., xi+1,...,xn) is an element of the subalgebra generated by the generators mentioned. It is well known that if C2 coincides with P2 or A2 then AutC2 = TAut C2 for the case of an arbitrary field K (see [13-17]). If K is a field of characteristic zero, then AutP3 * TAutP3 by [18-19] and Aut^3 * TAut A3 by [20-21].
An automorphism of an algebra Cn is said to be unitriangular if it has the form
9 :x; ^ xt + (x1 ,..., xt_1), (5)
where i = 1, ..., n, and ft(x-l, ...,x;_1) is an element of the subalgebra generated by the generators mentioned. Let TUn denote a subgroup of ofTAutCn which is generated by elementary uni-triangularautomorphisms of the form
4> : X; ^ X; + ,..., Xi_1),Xj ^ Xyfor i ^ j, (6)
where i,j = 1,...,n and f^ ,..., x;_1)is an element of the subalgebra generated by the generators mentioned. We assume that an element x1 is kept fixed under any unitriangularautomor-phism. Formally, if algebra contains free members, this element may have an image of the form x1+ a, where a is an element of K. Under such a definition, the group ofunitriangularau-tomorphisms differs inessentially from the group defined in the way indicated above, which the former contains as a subgroup. The following result will also be valid.
Theorem 11 (Roman'kov, Chirkov, Shevelin [22]). The group of tame automorphisms of the free Lie algebra Ln(the free associative algebraAn, the absolutely free algebra Fn, the algebra Pn of polynomials) of rank n > 4 over a field K of characteristic zero admits no faithful representation by matrices over any field.
More exactly, we establish in all these cases that the group of tame automorphisms contains a solvable subgroup of unitriangu-larautomorphismsTUn in which the commutant of every subgroup of finite index is not nilpotent. By a theorem of A. I. Malcev, this is impossible in matrix groups.
Similarly to TUn, we define a group TCn of triangular automorphisms which is generated by elementary triangular automorphisms like
4> : X;^ aXj + f(x1,...,x;_1), Xj^ Xj for i ^ j, (7) wherei,j = 1,...,n, and a is a nonzero element of K. The group TCn consists of all automorphisms of the form
0 : Xj ^ atxt + (x1 , ...,x;_!), (8)
wherei = 1,...,n, at is a nontrivial element of K, and (x-l , ...,x;_1)are elements of the subal-gebras generated by the generators specified.
Theorem 12
(a) (Sosnovskii [23]). For n >3 the group AutPn over a field K of characteristic zero is not linear.
(b) (Bardakov, Neschadim, Sosnovskii [24]).For n >3 the group Aut4„ over a field K of characteristic zero is not linear.
Theorem 13 (Roman'kov [25]). For algebras Pn, An, and Z^every finitely generated subgroup G of a group of triangular automorhisms admits a faithful representation by triangular matrices over K. Consequently, the group G is soluble. At the same time, every finitely generated subgroup H of a group of unitriangularau-tomorphisms admits a faithful representation by unitriangular matrices over K. Hence the group H is nilpotent.
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