ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

2. Algebra

1991 MS Classification: 20 Ivanov, Sergei, University of Illinois at Urbana-Champaign, U. S. A. On the Burnside problem for groups of even exponent

Recall that the notorious Burnside problem (posed in 1902, see [B]) asks whether any finitely generated group that satisfies the law x^{n} ≡ 1 (n is a fixed positive integer called the exponent of G) is necessarily finite. This is proven only for exponents n ≤ 4 and n = 6 (the case n = 3 is due to Burnside, n = 4 to Sanov, and n = 6 to Hall, see [MKS] ). A negative solution to the Burnside problem (for odd exponents) was given by Novikov and Adian [NA] in 1968 who proved the existence of infinite m-generator groups with m ≥ 2 of any odd exponent n ≥ 4381 (a simpler geometric solution to this problem for odd n > 10^{10} was later given by Ol’shanskii [Ol]; note that Adian [Ad] later improved on the original estimate to odd n ≥ 665). We remark that attempts to approach the Burnside problem via finite groups gave rise to a restricted version of the Burnside problem [M] that asks whether there exists a number f(m, n) so that the order of any finite m-generator group of exponent n is less than f(m, n). The existence of such a bound f(m, n) was proven for prime n by Kostrikin [K] and for n = p^{ℓ} with prime p by Zelmanov [Z1]-[Z2]. However, the Burnside problem for even exponents n without odd divisor ≥ 665, especially interesting for n = 2^{ℓ} ≫ 1, remained open. The principal difference between odd and even exponents in the Burnside problem can be illustrated by pointing out that, on the one hand, for every odd n ≫ 1 there are infinite 2-generator groups of exponent n all of whose proper subgroups are cyclic and, on the other hand, any 2-group the orders of whose abelian (or finite) subgroups are bounded is itself finite. A negative solution to the Burnside problem for even exponents n ≫ 1 given in speaker’s article [Iv] is based on the following inductive construction (which is analogous to Ol’shanskii’s construction [Ol] for odd n > 10^{10}). Let F_{m} be a free group over an alphabet A = {a_{1}^{±1}, …, a_{m}^{±1}}, m > 1, n ≥ 2^{48} and n be divisible by 2^{9} (from now on we impose these restrictions on m and n unless otherwise stated; note this estimate n ≥ 2^{48} has been improved on by Lysenok [L] to n ≥ 2^{13}). By induction on i, let B(m, n, 0) = F_{m} and, assuming that the group B(m, n, i − 1) with i ≥ 1 is already constructed as a quotient group of F_{m}, define A_{i} to be a shortest element of F_{m} (if any) the order of whose image in B(m, n, i − 1) (under the natural epimorphism ψ_{i−1} : F_{m} → B(m, n, i − 1)) is infinite. Then B(m, n, i) is constructed as a quotient group of B(m, n, i − 1) by the normal closure of ψ_{i−1}(A_{i}^{n}). Clearly, B(m, n, i) has a presentation of the form B(m, n, i) = ⟨ a_{1}, …, a_{m} ∥ A_{1}^{n}, …, A_{i−1}^{n}, A_{i}^{n} ⟩, where A_{1}^{n}, …, A_{i−1}^{n}, A_{i}^{n} are the defining relators of B(m, n, i). It is proven in [Iv] that for every i the word A_{i} does exist. Furthermore, it is shown that the direct limit B(m, n, ∞) of the groups B(m, n, i) as i → ∞ (obtained by imposing on F_{m} of relators A_{i}^{n} for all i = 1, 2, …) is exactly the free m-generator Burnside group B(m, n) of exponent n (B(m, n) is the quotient group F_{n}/F_{m}^{n}, where F_{m}^{n} is the subgroup generated by all nth powers of elements of F_{m}). The infiniteness of the group B(m, n) already follows from the existence of the word A_{i} for every i ≥ 1, since, otherwise, B(m, n) could be given by finitely many relators and so A_{i} would fail to exist for sufficiently large i. It is also shown that the word and conjugacy problems for the group B(m, n) are solvable (for odd n ≥ 665 it is known due to Novikov and Adian [Ad]). As a matter of fact, these decision problems are effectively reduced to the word problem for groups B(m, n, i) and it is shown that each B(m, n, i) satisfies a linear isoperimetric inequality and hence B(m, n, i) is a hyperbolic group in the sense of Gromov [G]. It is worth noting that the structure of finite subgroups of the groups B(m, n, i), B(m, n) is very complex when the exponent n is even and, in fact, finite subgroups of B(m, n, i), B(m, n) turn out to be so important in proofs of [Iv] that at least a third of article [Iv] is an investigation of their various properties and another third is a preparation of necessary techniques to conduct this investigation. The central result related to finite subgroups of the groups B(m, n, i), B(m, n) is the following: Let n = n_{1}n_{2}, where n_{1} is the maximal odd divisor of n. Then any finite subgroup G of B(m, n, i), B(m, n) is isomorphic to a subgroup of the direct product D(2n_{1})×D(2n_{2})^{ℓ} for some ℓ, where D(2k) denotes a dihedral group of order 2k (for odd n ≥ 665 it is known due to Novikov and Adian that all finite and abelian subgroups of B(m, n) are cyclic [Ad]). The natural question on whether this description is complete (i. e. every subgroup of D(2n_{1}) × D(2n_{2})^{ℓ} can actually be found in B(m, n)) is solved in the affirmative in a joint work with Ol’shanskii [IvOl1] by proving the following theorem: Let G be a finite 2-subgroup of B(m, n). Then the centralizer C_{B(m,n)}(G) of G in B(m, n) contains a subgroup B isomorphic to a free Burnside group B(∞, n) of infinite countable rank such that G ∩ B = {1}, whence ⟨G, B⟩ = G × B. (Since B(∞, n) obviously contains subgroups isomorphic to both D(2n_{1}) and D(2n_{2}) an embedding of D(2n_{1}) × D(2n_{2})^{ℓ} in B(m, n) becomes trivial.) Among other results on subgroups of B(m, n) proven in [IvOl1] are the following: The centralizer C_{B(m,n)}(S) of a subgroup S is infinite if and only if S is a locally finite 2-group. Any infinite locally finite subgroup L is contained in a unique maximal locally finite subgroup while any finite 2-subgroup is contained in continuously many pairwise nonisomorphic maximal locally finite subgroups. A clear complete description of infinite (maximal) locally finite subgroups of B(m, n) is also obtained. The machinery developed in [Iv] for solving the Burnside problem for even exponents has turned out to be useful for other problems. For example, Gromov [G] conjectured that for every nonelementary ( = non-cyclic-by-finite) hyperbolic group G there exists an n = n(G) such that the quotient G/G^{n} is infinite. This Gromov conjecture (along with other results on such quotients G/G^{n}) is proven in [IvOl2] by carrying out the machinery of [Iv] to nonelementary hyperbolic groups. References:

[Ad] S. I. Adian, The Burnside problems and identities in groups, Springer, 1979.

[B] W. Burnside, On unsettled question in the theory of discontinuous groups, Quart. J. Pure and Appl. Math. 33 (1902), 230–238.

[G] M. Gromov, Hyperbolic groups, in Essays in Group Theory; S. M. Gersten, ed., M. S. R. I. Pub. 8, Springer, 1987, 75–263

[Iv] S. V. Ivanov, The free Burnside groups of sufficiently large exponents, Intern. J. Algebra Comp. 4 (1994), 1–308.

[IvOl1] S. V. Ivanov and A. Yu. Ol’shanskii, On finite and locally finite subgroups of free Burnside groups of large even exponents, J. Algebra 195 (1997), 241–284.

[IvOl2] S. V. Ivanov and A. Yu. Ol’shanskii, Hyperbolic groups and their quotients of bounded exponents, Trans. Amer. Math. Soc. 348 (1996), 2091–2138.

[K] A. I. Kostrikin, On the Burnside problem, Math. USSR Izv. 23 (1959), 3–34.

[L] I. G. Lysenok, Infinite Burnside groups of even period, Math. Ross. Izv. 60 (1996), 3–224.

[M] W. Magnus, A connection between the Baker-Hausdorff formula and a problem of Burnside, Ann. Math. 52 (1950), 11–26; 57 (1953), 606.

[MKS] W. Magnus, J. Karras, and D. Solitar, Combinatorial group theory, Interscience Pub., John Wiley and Sons, 1966.

[NA] P. S. Novikov and S. I. Adian, On infinite periodic groups I, II, III, Math. USSR Izv. 32 (1968), 212–244, 251–524, 709–731.

[Ol] A. Yu. Ol’shanskii, On the Novikov-Adian theorem, Math. USSR Sb. 118 (1982), 203–235.

[Z1] E. I. Zelmanov, Solution of the restricted Burnside problem for groups of odd exponent, Math. USSR Izv. 36 (1991), 41–60.

[Z2] E. I. Zelmanov, A solution of the restricted Burnside problem for 2groups, Math. USSR Sb. 72 (1992), 543-565.