In this paper, we study the action of a countable group Γ on the space of orders on the group. In particular, we are concerned with the invariant probability measures on this space, known as invariant random orders. We show that for any countable group, the space of random invariant orders is rich enough to contain an isomorphic copy of any free ergodic action, and characterize the non-free actions realizable. We prove a Glasner–Weiss dichotomy regarding the simplex of invariant random orders. We also show that the invariant partial order on SL3(Z) corresponding to the semigroup generated by the standard unipotents cannot be extended to an invariant random total order. We thus provide the first example for a partial order (deterministic or random) that cannot be randomly extended.
Classification :
20F60, 37A15
Mots-clés :
orders on groups, random orders, amenability
Affiliations des auteurs :
Yair Glasner
1
;
Yuqing Frank Lin
1
;
Tom Meyerovitch
1
1
Ben-Gurion University of the Negev, Be’er Sheva, Israel
Yair Glasner; Yuqing Frank Lin; Tom Meyerovitch. Extensions of invariant random orders on groups. Groups, geometry, and dynamics, Tome 18 (2024) no. 4, pp. 1377-1401. doi: 10.4171/ggd/785
@article{10_4171_ggd_785,
author = {Yair Glasner and Yuqing Frank Lin and Tom Meyerovitch},
title = {Extensions of invariant random orders on groups},
journal = {Groups, geometry, and dynamics},
pages = {1377--1401},
year = {2024},
volume = {18},
number = {4},
doi = {10.4171/ggd/785},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/785/}
}
TY - JOUR
AU - Yair Glasner
AU - Yuqing Frank Lin
AU - Tom Meyerovitch
TI - Extensions of invariant random orders on groups
JO - Groups, geometry, and dynamics
PY - 2024
SP - 1377
EP - 1401
VL - 18
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/785/
DO - 10.4171/ggd/785
ID - 10_4171_ggd_785
ER -
%0 Journal Article
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%A Yuqing Frank Lin
%A Tom Meyerovitch
%T Extensions of invariant random orders on groups
%J Groups, geometry, and dynamics
%D 2024
%P 1377-1401
%V 18
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/785/
%R 10.4171/ggd/785
%F 10_4171_ggd_785
