Let \(\Delta(G)\) denote the augmentation ideal of the integral group ring \(ZG\) and \(\Delta^ n(G)\) its \(i\)th power. Let \(\Delta^{(1)}(G)=\Delta(G)\) and \(\Delta^{(i)}(G)=(\Delta^{(i- 1)}(G),\Delta(G))ZG\) for \(i>1\). Here \((A,B)ZG\) denotes the ideal of \(ZG\) generated by all Lie products \((a,b)=ab-ba\), with \(a\in A\), \(b\in B\). If \(D_ n(G)=G\cap(1+\Delta^ n(G))\) and \(D_{(n)}(G)=G\cap(1+\Delta^{(n)}(G))\), we call \(D_ n(G)\) and \(D_{(n)}(G)\) the \(n\)th dimension and the \(n\)th Lie dimension subgroup of \(G\) respectively. If \(\gamma_ n(G)\) denotes the \(n\)th term of the lower central series for \(G\) then \(\gamma_ n(G)\subseteq D_{(n)}(G)\subseteq D_ n(G)\).
The question whether \(D_ n(G)=\gamma_ n(G)\), known as the dimension subgroup conjecture, is known to be true if \(n\leq 3\), and was shown false for \(n=4\) by Rips in 1972. In 1987 Gupta showed, via examples of metabelian groups, that the conjecture is false for \(n\geq 4\). By a result of Sandling the Lie dimension subgroup conjecture, \(D_{(n)}(G)=\gamma_ n(G)\), was known to be true if \(n\leq 6\). In this paper the authors settle the Lie dimension subgroup conjecture by showing that it is false if \(n\geq 9\). The authors construct their counter examples to the conjecture by inserting the Gupta group as a subgroup in a certain group of larger nilpotency class. The construction requires involved commutator arguments.
If \(\Delta^{[1]}(G)=\Delta(G)\) and for \(i\geq 1\), \(\Delta^{[i+1]}(G)=(\Delta^{[i]}(G),\Delta(G))\), and \(D_{[n]}(G)=G\cap(1+\Delta^{[n]}(G)ZG)\) then in 1983 Gupta and Levin showed that \(\gamma_ n(G)\subseteq D_{[n]}(G)\subseteq D_{(n)}(G)\subseteq D_ n(G)\) for all \(n\). The authors also show in this paper that for \(n\geq 14\) that \(D_{[n]}(G)\neq \gamma_ n(G)\).