The celebrated construction due to Golod and Shafarevich gives an infinite dimensional algebra satisfying some prescribed relations. Some of its corollaries include the construction of a nil, finitely generated algebra which is not nilpotent, and this led to a straightforward negative answer to the general Burnside problem. The so-called Golod-Shafarevich algebras have been extensively studied but nevertheless there are many open problems concerning their structure and growth.
The paper under review constructs an algebra in the following way. Assume \(K\) is an algebraically closed field, and \(A\) the free associative algebra freely generated over \(K\) by two elements \(x\) and \(y\). Take homogeneous elements \(f_1,f_2,\ldots\), in \(A\) and let \(I\) be the ideal in \(A\) generated by these elements. Suppose there are \(r_n\) of these elements whose degrees are \(>2^n\) and \(\leq 2^{n+1}\), and moreover for \(i<8\), all \(r_i=0\). Let \(Y=\{n\mid r_n\neq 0\}\). Impose further the following two conditions.
1. There is no \(f_i\) such that \(\deg f_i=k\in[2^n-2^{n-3},2^n+2^{n-2}]\) for some positive integer \(n\).
2. If \(m,n\in Y\cup\{0\}\) and \(m<n\) then \(2^{3n+4}r_m^{33}<r_n<2^{2^{n-m-3}}\) and \(r_n<2^{2^{n/2-4}}\).
Then there exists an algebra \(R\) which is a homomorphic image of \(A/I\) such that:
1. \(R\) is graded and infinite dimensional.
2. If for each \(n\) one denotes by \(k\) the largest positive integer in \(Y\) with \(k\leq 2\log n\) then \(\dim R_n\leq 8n^4r_k^{33}\) (here \(R_n\) stands for the span of all elements of degree \(\leq n\) in \(R\)).
3. If \(j\in Y\) and \(j\leq\log n\) then \(r_j^4\leq 2\dim R_n\).
The above example provides algebras with subexponential growth satisfying given relations. An example of a nil algebra with neither polynomial nor exponential growth is also given. Further consequences of the above construction are also given.