\((2,3,k)\)-generated groups of large rank
Let \(k\) be a positive integer. The triangle group \(\Delta(2,3,k)\) is the group given by the presentation \(\langle X,Y\mid X^2=Y^3=(XY)^k=1\rangle\). A group is said to be \((2,3,k)\)-generated if it is a non-trivial image of \(\Delta(2,3,k)\). It is known that \(\Delta(2,3,k)\) is finite precisely when \(k\leq 5\) and the group \(\Delta(2,3,6)\) is an extension of a free Abelian group of rank 2 by the cyclic group \(C_6\). In this paper, the author proves that, for any \(k\geq 7\), there exist integers \(n_k\) and \(a_k\) such that, if a ring \(R\) is generated by a set of \(m\) elements \(t_1,\ldots,t_m\), where \(2t_1-t_1^2\) is a unit of finite multiplicative order, then the elementary linear group \(E_n(R)\) is \((2,3,k)\)-generated for all \(n\geq n_k+ma_k\).
scientific article; zbMATH DE number 1394461