2020/01/28 by J. Alonso, Alonso, J., Y. Antolín +5
Mathematics · #20E08 #20F60 #20F67 #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG) #math.GR #math.MG #msc:20E08 #msc:20F60 #msc:20F67
paper · pdf · doi:10.48550/arxiv.2001.10286
32 pages
arxiv created 2022/01/28 · arxiv updated 2022/01/31
We study the geometry of positive cones of left-invariant total orders (left-order, for short) in finitely generated groups. We introduce the Hucha property and the \texitPrieto property for left-orderable groups. The first one means that in any left-order the corresponding positive cone is not coarsely connected, and the second one that in any left-order the corresponding positive cone is coarsely connected. We show that all left-orderable free products have the Hucha property, and that the Hucha property is stable under certain free products with amalgamatation over Prieto subgroups. As an application we show that non-abelian limit groups in the sense of Z. Sela (e.g. free groups, fundamental group of hyperbolic surfaces, doubles of free groups and others) and non-abelian finitely generated subgroups of free ℚ-groups in the sense of G. Baumslag have the Hucha property. In particular, this implies that these groups have empty BNS-invariant Σ1 and that they don't have finitely generated positive cones.