2026/07/28 by Lucía Asencio-Martín, John R. Britnell, Andrew Duncan +2
Mathematics · #math.GR #msc:08A50 #msc:20F05 #msc:20F06 #msc:20F10 #msc:20F65
24 pages, 6 figures
arxiv created 2026/07/28 · arxiv updated 2026/07/30
Let G be an automatic group with associated regular language L. We describe a procedure for constructing an automaton which recognises elements of a given submonoid or rational subset K of G. This builds on work of Kharlampovich, Miasnikov and Weil, on the case where K is a subgroup of G. Our construction succeeds, after sufficiently many iterations, whenever K satisfies a certain convexity property, which we call L-proximity. We show how to test whether the construction is complete in the case that K is a submonoid; we have no such test for the general case of a rational subset K. We focus particularly on the case of a surface group G of genus g>1, where L is the language of geodesic words in the standard generators. We use small cancellation theory to obtain a method for constructing L-recognisable submonoids of G.