2007/06/22 by Stephen J. Pride, Stephen J Pride, Pride, Stephen J
Mathematics · #20E26 #20F05 (Primary) 20F10 #57M07 (Secondary) #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20E26 #msc:20F05 #msc:20F10 #msc:57M07
paper · pdf · doi:10.48550/arxiv.0706.3338
arxiv created 2007/06/22 · openalex publication_date 2007/06/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A relative one-relator presentation has the form P = < X,H ; R > where X is a set, H is a group, and R is a group word on X and H. We show that if the group word on X obtained from R by deleting all the terms from H has what we call the unique max-min property, then the group defined by P is residually finite if and only if H is residually finite (Theorem 1). We apply this to obtain new results concerning the residual finiteness of (ordinary) one-relator groups (Theorem 4). We also obtain results concerning the conjugacy problem for one-relator groups (Theorem 5), and results concerning the relative asphericity of presentations of the form P (Theorem 6).