Exponents of Subvarieties of Upper Triangular Matrices over Arbitrary Fields are Integral

Partially supported by grant RFFI 98-01-01020.Let Uc be the variety of associative algebras generated by the algebra of all upper triangular matrices, the field being arbitrary. We prove that the upper exponent of any subvariety V ⊂ Uc coincides with the lower exponent and is an integer

On growth of Lie algebras, generalized partitions, and analytic functions

AbstractIn this paper we discuss some recent results on two different types of growth of Lie algebras that lead to some combinatorial problems. First, we study the growth of finitely generated Lie algebras (Sections 1–4). This problem leads to a study of generalized partitions. Recently the author has suggested a series of q-dimensions of algebras Dimq,q∈N which includes, as first terms, dimensions of vector spaces, Gelfand–Kirillov dimensions, and superdimensions. These dimensions enabled us to describe the change of a growth in transition from a Lie algebra to its universal enveloping algebra. In fact, this is a result on some generalized partitions. In this paper we give some results on asymptotics for those generalized partitions. As a main application, we obtain an asymptotical result for the growth of free polynilpotent finitely generated Lie algebras. As a corollary, we specify the asymptotic growth of lower central series ranks for free polynilpotent finitely generated groups. We essentially use Hilbert–Poincaré series and some facts on growth of complex functions which are analytic in the unit circle. By growth of such functions we mean their growth when the variable tends to 1. Also we discuss for all levels q=2,3,… what numbers α>0 can be a q-dimension of some Lie (associative) algebra. Second, we discuss a ‘codimension growth’ for varieties of Lie algebras (Sections 5 and 6). It is useful to consider some exponential generating functions called complexity functions. Those functions are entire functions of a complex variable provided the varieties of Lie algebras are nontrivial. We compute the complexity functions for some varieties. The growth of a complexity function for an arbitrary polynilpotent variety is evaluated. Here we need to study the connection between the growth of a fast increasing entire function and the behavior of its Taylor coefficients. As a result we obtain a result for the asymptotics of the codimension growth of a polynilpotent variety of Lie algebras. Also we obtain an upper bound for a growth of an arbitrary nontrivial variety of Lie algebras

Lie Algebras and Growth in Branch Groups

We compute the structure of the Lie algebras associated to two examples of branch groups, and show that one has finite width while the other, the ``Gupta-Sidki group'', has unbounded width. This answers a question by Sidki. More precisely, the Lie algebra of the Gupta-Sidki group has Gelfand-Kirillov dimension $\log3/\log(1+\sqrt2)$. We then draw a general result relating the growth of a branch group, of its Lie algebra, of its graded group ring, and of a natural homogeneous space we call "parabolic space", namely the quotient of the group by the stabilizer of an infinite ray. The growth of the group is bounded from below by the growth of its graded group ring, which connects to the growth of the Lie algebra by a product-sum formula, and the growth of the parabolic space is bounded from below by the growth of the Lie algebra. Finally we use this information to explicitly describe the normal subgroups of the "Grigorchuk group". All normal subgroups are characteristic, and the number of normal subgroups of index $2^n$ is odd and is asymptotically $n^{\log_2(3)}$