Word values in \(p\)-adic and adelic groups. (Q2865928)

The authors of this paper study word values and word width in simple algebraic groups over the \(p\)-adic integers and over general local rings.NEWLINENEWLINE Some of their results are the following:NEWLINENEWLINE \(\bullet\) Suppose that \(G\) is a semisimple, simply connected, algebraic group over \(\mathbb Q\), and that \(w_1,w_2,w_3\) are non-trivial words. If \(p\) is large enough, then \(w_1(G(\mathbb Z_p))\cdot w_2(G(\mathbb Z_p))\cdot w_3(G(\mathbb Z_p))=G(\mathbb Z_p)\). In particular, \(w(G(\mathbb Z_p))^3=G(\mathbb Z_p)\) for \(w\neq 1\) and sufficiently large \(p\).NEWLINENEWLINE \(\bullet\) Let \(w_1,w_2\) be non-trivial words, and let \(G\) be a semisimple simply connected algebraic group over \(\mathbb Q\). If \(p\) is large enough, then \(w_1(G(\mathbb Z_p))\cdot w_2(G(\mathbb Z_p))\supset G(\mathbb Z_p)\setminus(Z(G)\cdot G^1(\mathbb Z_p))\).NEWLINENEWLINE \(\bullet\) If \(O\) is a local ring whose residue field has more than \(n+1\) elements, then every element of \(\mathrm{SL}_n(O)\) that is not scalar modulo \(\mathfrak m\) is a commutator.NEWLINENEWLINE \(\bullet\) Let \(O\) be a local ring with finite residue field \(K\), and let \(n\) be a proper divisor of \(|K|-1\). Then every element in \(\mathrm{PSL}_n(O)\) is a commutator.