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УДК 512.54
The Linearity Problem for the Unitriangular Automorphism Groups of Free Groups
Vitaly A. Roman'kov*
Institute of Mathematics and Information Technologies, Omsk F.M. Dostoevsky State University, Mira, 55-A, Omsk, 644077. Omsk State Technical University, Mira, 11, Omsk, 644050, Russia
Received 10.08.2013, received in revised form 16.09.2013, accepted 20.10.2013 We prove that the unitriangular automorphism group of a free group of rank n has a faithful representation by matrices over a field, or in other words, it is a linear group, if and only if n < 3. Thus, we have completed a description of relatively free groups with linear the unitriangular automorphism groups. This description was initiated by Erofeev and the author in [1], where proper varieties of groups have been considered.
Keywords: free group, unitriangular automorphism, linearity.
Introduction
For each positive integer n, let Fn be a free group of rank n with basis (in other words, free generating set) |/i,...,/n}. For any m < n Fm is considered as subgroup gp(/i,...,/m) of Fn. For any variety of groups G, let G (Fn) denote the verbal subgroup of Fn corresponding to G. Let Gn = Fn/G(Fn). Then Gn is a relatively free group of rank n in the variety G. By basis of Gn we mean a subset S such that every map of S into Gn extends, uniquely, to an endomorphism of Gn. Write fi = /iG(Fn) for i = 1, ...,n. Then /1; ...,/n is a basis of Gn.
Let G be a variety of groups. Let Gn be the relatively free group corresponding to G with basis {/1,..., /n}. For any m < n Gm is considered as subgroup gp(/1,..., /m) of Gn. An automorphism f of Gn is called unitriangular (w.r.t. the given basis) if f is defined by a map of the form:
f : fi ^ fi,fi ^ uifi для i = 2,..., n, (1)
where u = uj(fi,..., fi-i) is an element of Gj_i. Every tuple of elements (u2, ...,un) with this condition defines, uniquely, automorphism of Gn. Let Un be subgroup consisting of all unitriangular (w.r.t. a given basis) automorphisms of Gn. Then it is called the unitriangular automorphism group of Gn. As abstract group Un does not depend of a basis.
The question of linearity of Un for an arbitrary proper variety G has been studied by Erofeev and the author in [1]. All cases of linearity of Gn have been described. The following Section 1 contains this description. Also, we observe some relative results on linearity for relatively free groups and algebras.
In this paper we study the only open after [1] case when G is the variety of all groups. Our main result is given by the following theorem.
Theorem 1. The group Un of unitriangular automorphisms of the free group Fn of rank n is linear if and only if n ^ 3.
* [email protected] © Siberian Federal University. All rights reserved
Hence, we complete a description of all cases when the unitriangular automorphism group Un, corresponding to an arbitrary variety of groups G, including the variety of all groups, is linear.
1. Some results on linearity
We observe some results concerning the linearity of the automorphism groups and their subgroups of relatively free groups and algebras. Recall that group G is said to be virtually nilpotent if it has a nilpotent subgroup of finite index.
The linearity of Aut(F2) follows by [2] from the linearity of the 4—string braid group B4, which is due to Krammer [3]. Bigelow [4] and also Krammer [5] determined that the braid group Bn is linear for every n. Formanek and Procesi in [6] have demonstrated that Aut(Fn) is not linear for n ^ 3.
Auslander and Baumslag [7] determined that for every finitely generated virtually nilpotent group G the automorphism group Aut(G) is linear. Moreover, Aut(G) has a faithful matrix representation over the integers Z. In particular, for every relatively free virtually nilpotent group Gn, the automorphism group Aut(Gn) is linear over Z.
Olshanskii [8] proved for any relatively free group Gn, which is not virtually nilpotent and is not free, that the automorphism group Aut(Gn) is not linear. His approach does not give an information on the linearity of the unitriangular automorphism groups Un for such relatively free groups Gn.
Erofeev and the author [1] proved for every proper variety of groups G that the unitriangular automorphism group Un is linear if and only if the relatively free group Gn-1 is virtually nilpotent. More exactly (for n > 3): if Gn-1 is virtually nilpotent, then Un admits a faithful matrix representation over integers Z. It was also shown in [1] that if n > 3 and Gn-1 is nilpotent then Un is nilpotent too.
Now let Cn be an arbitrary relatively free algebra of rank n with set of free generating elements {x1, ...,xn}. For m < n Cm can be considered as subalgebra of Cn generated by x1, ...,xm. An automorphism ф of Cn is called unitriangular w.r.t. the given set of free generating elements if it is defined by map of the form:
ф : X1 ^ x1,xi ^ xi + ui для i = 2,..., n, (2)
where ui = ui(x1,..., xi-1) belongs to Ci-1. Let Un denote a subgroup of the automorphism group Aut(Cn) of Cn, consisting of all unitriangular automorphisms. As abstract group Un does not depend from a chosen set of free generating elements of Cn.
The author, Chirkov and Shevelin [9] proved that, for a free Lie (free associative, absolutely free, polynomial) algebra Cn of rank n ^ 4 over a field of zero characteristic, the unitriangular automorphism group Un is not linear. Then the following papers [10,11] presented descriptions of the hypercentral series of groups Un corresponding to polynomial and free metabelian Lie algebras, respectively. By these results Un are not linear for n > 3. By [12], for n > 3, the unitriangular automorphism group Un is not linear in case of polynomial algebra and in case of free associative algebra. By [13] for each relatively free algebra Cn the group Un is locally nilpotent, thus it is linear.
2. The method of Formanek and Procesi
Let G be any group, and let H(G) denote the following HNN-extension of G x G:
H(G) =< G x G, t : t(g, g)t-1 = (1, g), g e G > . (3)
Theorem 2.1 (Formanek, Procesi [6]). 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.
Theorem 2.2 (Brendle, Hamidi-Tehrani [14]). 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-abelian-by-finite. Then H(G)/N is not linear.
In [14] a group of the type described in Theorem 2.2 is called a Formanek and Procesi group, or FP-group for short.
3. Proof of Theorem 1
2.1. For n ^ 3, Un is linear.
Proof. Since U1 is trivial and U2 is infinite cyclic the statement is obvious for n =1, 2. Let n = 3. By [1] U3 is generated by automorphisms A2j1, A3j1, A3,2. Recall that Ai,j maps /i to /j/i, and fixes all other basic elements. This is applicable for any group Un. The automorphisms A3j1 A3,2 generate in U3 a normal free subgroup F2. The automorphism A2j1 acts as follows:
A2,1 A3,1 A2,1 = A3,1, A2jA3,2A2,1 = A3,1 A3,2. (4)
Now we'll show that U3 is isomorphic to a subgroup of Aut(F2). Let t1 and t2 denote inner automorphisms of F2 corresponding to /1 and /2 respectively. This means that any element g of F2 maps by ri(i = 1,2) to /i-1 g/i. Let a2j1 G Aut(F2) fixes /1 and maps /2 to /1/2. Obviously, F2 = gp(r1,r2) is a free group of rank 2. It is a normal subgroup of V3 = gp(r1, t2, a2j1). A quotient V3/F2 is the infinite cyclic generated by the image of a2j1. The corresponding action is determined by:
а2ДriCT2,i = Ti, ст2,1т2а2,1 = TiT2. (5)
Thus, U3 and V3 are both infinite cyclic extensions of F2. By (4) and (5) we conclude that
a : U3 ^ V3 defined as: -10л
a : Аз,^ ^ Tj для j = 1, 2, A2,i ^ 02,i, (6)
is isomorphism. Since V3 is a subgroup of Aut(F2), which is linear by [2] and [3], U3 is also linear. □
2.2. For n ^ 4, Un is not linear.
Proof. For n > m, Un has a subgroup that is isomorphic to Um. Elements of this subgroup act naturally to fi,..., fm and fix elements fm+i,..., fn. So, we just have to prove that U4 is not linear.
By Theorem 2.2 it will be enough to find a subgroup H of U4 that is isomorphic to a quotient H(F2)/N, where H(F2) is given by (3), such that the image of G x {1} in H(G)/N is not nilpotent-by-abelian-by-finite.
There are two commuting elementwise subgroups of U4 each of them is isomorphic to F2. Namely, there are gp(A3ji, A3,2) and gp(A4ji, A4,2). Consider them as two copies of F2 via isomorphism defined by map A3ji ^ A4ii,A3j2 ^ A4,2. Thus we have a subgroup F2 x F2 of U4. Easily to check that:
Л4,3 A3,j A4,j A4,3 = A3,j,
A4 3A3,jA4,jA4,3 = A3,j, j = 1, 2. (7)
By (3) and (7) we conclude that H is a homomorphic image of H(F2) such that the subgroup F2 x F2 of H(F2) maps isomorphically to the just constructed subgroup of the same type of U4. The image of t is A4 3. Hence, H ^ H(F2)/N, where N is the kernel of this homomorphism. By Theorem 2.2 H, and so U4, is not linear. □
Remark 1. In fact we proved that subgroup W4 = gp(A3,j, A4jl : j = 1, 2; l = 1, 2, 3) of U4 is not linear. In [1] we noted that [Ai,j, Aj,k] = Ai k. Here commutator [g,/] means g/g-1/-1. Any group Un is generated by the elements Ai,j, for j < i ^ n (see [1]). It follows that W4 is the derived subgroup U4 of U4. Hence, we proved that the derived subgroup (the second member 72U4 of the low central series) of U4 is not linear. This subgroup y2U4 has also characterized in U4 as the stabilizer of /1. In general case, for n ^ 4, member Yn-2 Un coincides with the elementwise stabilizer of {/1,..., /n-3}. Easily to see that the derived subgroup U4 can be embedded into Yn-2Un. Hence, for every n ^ 4, a member Yn-2Un of the low central series of Un is not linear. This statement is more strong than the statement of Theorem 1 about nonlinearity of Un for n ^ 4.
Remark 2. In [15] an explicit faithful representation of Aut(F2) in GL12(Z[t±1,q±1]) is given. Here Z[t±1,q±1] is a Laurent polynomial ring. Hence, U3 has a faithful matrix representation over Z[t±1,q±1 ].
Also note that U3 can be presented by < a, b : [[a, b], b]] = 1 >, where a corresponds to A3 2, and b corresponds to A2j1. By terminology of [17] U3 is the first non commutative member in a series of Hydra groups =< a, b : [...[[a, b], b],..., b] = 1 >,k ^ 1, where commutator has k entries of b. In general, Hydra groups were introduced in [16]. It was shown in [17] that Hydra groups of such form are residually torsion-free nilpotent. It seems interesting to study their linearity. Remark 3. By [18] a group G is called locally graded if every nontrivial finitely generated subgroup of G has a proper subgroup of finite index. This class contains, for example, all locally solvable and all residually finite groups. Let G2 be the relatively free group of rank 2 in the variety var(G) generated by G. Suppose that the derived subgroup G2 is finitely generated. Then by [18] G is virtually nilpotent.
Let G be a variety consisting of locally graded groups. Obviously G is a proper variety. Suppose that U3 is linear. Then every group Un is linear. Indeed, by [1] G2 is virtually nilpotent. It follows that G2 is finitely generated. Any group Gn-1 generates a subvariety var(Gn-1) of G. The relatively free group of rank 2 in this subvariety is a homomorphic image of G2, and so has finitely generated derived subgroup. Then by [18] Gn-1 is virtually nilpotent. It follows by [1] that Un is linear. Thus, the linearity of U3 implies the linearity of Un for every n ^ 4. Moreover, G should be virtually nilpotent.
We see by Theorem 1 that just presented statement, that the linearity of U3 implies the linearity of Un for all n > 4, is not true for the variety of all groups. Likely, it is also non-true for some proper varieties of groups. As candidates to such varieties we can consider the varieties of groups generated by the famous Golod groups. We conjecture that for every m > 3 there is a variety Gm such that the groups Un are linear if and only if n < m.
The investigation was supported by The Ministry of Education and Science of Russian Federation, projects Ц.В37.21.0359/0859, and by the RFBR, project 13-01-00239.
References
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Проблема линейности групп унитреугольных автоморфизмов свободных групп
Виталий А. Романьков
Доказано, что группа унитреугольных автоморфизмов свободной группы 'ранга n допускает точное представление матрицами над полем тогда и только тогда, когда n < 3. Таким образом завершено описание относительно свободных групп, группы унитреугольных автоморфизмов которых линейны, начатое работой С. Ю. Ерофеева и автора [1] , где рассмотрены все относительно свободные группы собственных многообразий групп.
Ключевые слова: свободная группа, унитреугольный автоморфизм, линейность.
