On the geometry of positive cones in finitely generated groups

Abstract

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 Prieto property for left-orderable groups. We say that a group has the Hucha property if in any left-order the corresponding positive cone is not coarsely connected, and the Prieto property if in any left-order the corresponding positive cone is coarsely connected. We show that all leftorderable 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 Σ 1 and that they do not have finitely generated positive cones.

Regular left-orders on groups

“…This formalized and strengthened an argument sketched by Calegari [8]. Finally, Alonso, Antolín, Brum and Rivas [1] proved that non-abelian limit groups (in particular, free groups and surface groups) do not admit coarsely connected positive cones, and therefore do not admit regular positive cones.…”

“…If P H \ P 1 6 D ;. Then there exists p 1 ; p 2 2 P and h 2 H such that p 1 h D p 1 2 , meaning that h D p 1 1 p 1 2 , so H is not disjoint from P 1 , a contradiction. Similarly, if P H \ H is non-empty, then there exists h 1 ; h 2 2 H and p 2 P such that ph 1 D h 2 , meaning that p D h 2 h 1 1 , which implies that P and H are not disjoint, a contradiction.…”

“…Let us first prove that P Â .L/. Let g 2 P , and assume that g D .q 1 n 1 q 1 1 /.q 2 n 2 q 1 2 / .q m n m q 1 m /p with q i ; n i ; p as in (1). Let…”

“…In previous works, the authors have given examples of groups not admitting regular left-orders [1,31]. In this paper we focus instead on constructing regular leftorders.…”

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