Generating pairs of projective special linear groups that fail to lift

Abstract

The following problem was originally posed by B.H. Neumann and H. Neumann. Suppose that a group $G$ can be generated by $n$ elements and that $H$ is a homomorphic image of $G$. Does there exist, for every generating $n$-tuple $(h_1,\ldots, h_n)$ of $H$, a homomorphism $\vartheta \colon G \to H$ and a generating $n$-tuple $(g_1,\ldots,g_n)$ of $G$ such that $(g_1^\vartheta,\ldots,g_n^\vartheta) = (h_1,\ldots,h_n)$? M.J. Dunwoody gave a negative answer to this question, by means of a carefully engineered construction of an explicit pair of soluble groups. Via a new approach we produce, for $n = 2$, infinitely many pairs of groups $(G,H)$ that are negative examples to the Neumanns' problem. These new examples are easily described: $G$ is a free product of two suitable finite cyclic groups, such as $C_2 \ast C_3$, and $H$ is a suitable finite projective special linear group, such as $\mathrm{PSL}(2,p)$ for a prime $p \ge 5$. A small modification yields the first negative examples $(G,H)$ with $H$ infinite.