We construct explicit finite generating sets for the stabilizers in Thompson’s group F of rational points of a unit interval or the Cantor set. Our technique is based on the Reidemeister-Schreier procedure in the context of Schreier graphs of such stabilizers in F. It is well known that the stabilizers of dyadic rational points are isomorphic to F×F and can thus be generated by 4 explicit elements. We show that the stabilizer of every non-dyadic rational point b∈(0,1) is generated by 5 elements that are explicitly calculated as words in generators x0,x1 of F that depend on the binary expansion of b. We also provide an alternative simple proof that the stabilizers of all rational points are finitely presented.
Krystofer Baker; Dmytro Savchuk. Explicit generators for the stabilizers of rational points in Thompson’s group $F$. Groups, geometry, and dynamics, Tome 19 (2025) no. 2, pp. 617-636. doi: 10.4171/ggd/890
@article{10_4171_ggd_890,
author = {Krystofer Baker and Dmytro Savchuk},
title = {Explicit generators for the stabilizers of rational points {in~Thompson{\textquoteright}s} group $F$},
journal = {Groups, geometry, and dynamics},
pages = {617--636},
year = {2025},
volume = {19},
number = {2},
doi = {10.4171/ggd/890},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/890/}
}
TY - JOUR
AU - Krystofer Baker
AU - Dmytro Savchuk
TI - Explicit generators for the stabilizers of rational points in Thompson’s group $F$
JO - Groups, geometry, and dynamics
PY - 2025
SP - 617
EP - 636
VL - 19
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/890/
DO - 10.4171/ggd/890
ID - 10_4171_ggd_890
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%A Dmytro Savchuk
%T Explicit generators for the stabilizers of rational points in Thompson’s group $F$
%J Groups, geometry, and dynamics
%D 2025
%P 617-636
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%N 2
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/890/
%R 10.4171/ggd/890
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