Topology on the Spaces Of Orderings of Groups
By Adam S. Sikora. 2001
110 citations recorded by the source corpus.
A natural topology on the space of left orderings of an arbitrary semi‐group is introduced here. This space is proved to be compact, and for free abelian groups it is shown to be homeomorphic to the Cantor set. An application of this result is a new proof of the existence of universal Gröbner bases. 2000 Mathematics Subject Classification 06F15, 13P10 (primary), 06F05, 20F60 (secondary).
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/cba444001552b46fccb4d25e7035529a00a21afb
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Foliations, orders, representations, L-spaces and graph manifolds (2014, unknown): “ring owith [αr(o)] = [α] arising from the short exact sequence 1 → hhαii → π1(M) → π1(M(α)) → 1. On the other hand if r≥ 3 and [α] is irrational then we use the topology on LO(π1(M)) defined by Sikora [Si]. Recall that a left-order ois determined by its positive cone P(o) ⊂ π1(M). Given x∈ π1(M), let”
- Amenable groups that act on the line (2006, unknown): “Sikora [12], but the action of Γ on this space does not seem to have appeared previously in the literature.”
- Foundations of Garside Theory (2013, background): “By contrast, forn > 2, if G is a countable free Abelian group of rank at least two, the spaceLO(G) (which coincides withO(G)) is infinite, homeomorphic to the Cantor set [207].”
- On the dynamics of (left) orderable groups (2007, methodology): “By applying Sikora’s theorem as above, we conclude that, in this case, O(Γ) is a Cantor set.”
- Ordered Groups and Topology (2015, background): “To our knowledge, this definition of LO(G) first appeared in [101].”
- Toric geometry and the Semple–Nash modification (2009, background): “It is shown in [28] that the closed subset of Z R(M) consisting of orders on M, with the induced topology, is homeomorphic to a Cantor set when the rank of the lattice M is ≥ 2.”
- GROUP ACTIONS ON 1-MANIFOLDS: A LIST OF VERY CONCRETE OPEN QUESTIONS (2017, methodology): “In this view, a useful idea independently introduced by Ghys [54] and Sikora [126] consists in endowing the space LO(G) of all left-orders of a given left-orderable group G with the Chabauty topology.”