Generating pairs of projective special linear groups that fail to lift

2019/08/15 by Boschheidgen, Jan, Klopsch, Benjamin, Thillaisundaram, Anitha
#20E06 #20F05 #20G40 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1908.05456

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 (h₁,…, h_n) of H, a homomorphism ϑ \colon G → H and a generating n-tuple (g₁,…,g_n) of G such that (g₁^ϑ,…,g_n^ϑ) = (h₁,…,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 C₂ ∗ C₃, and H is a suitable finite projective special linear group, such as PSL(2,p) for a prime p ≥ 5. A small modification yields the first negative examples (G,H) with H infinite.

Related

• Solving linear equations over finitely generated abelian groups
• Invariable generation of certain groups of piecewise projective homeomorphisms of the real line
• On the Hanna Neumann Conjecture
• The generating pairs of the 2-transitive groups
• Groups with no Parametric Galois Extension