Catalog · 10.5281/zenodo.23452788
The small index property for the random poset
- Martino Lupini · University of Bologna
18 October 2026 · CC BY 4.0 · latest of 2 versions
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The small index property for the random poset
Martino Lupini
Abstract
Let P be the countable universal homogeneous poset. We prove that every abstract subgroup H ≤ Aut(P) with [Aut(P):H] < 2^{ℵ₀} contains the pointwise stabilizer of a finite subset of P. Thus Aut(P) has the small index property, answering the question recorded as open by Barbina and Macpherson in 2005. No regularity assumption on H is required. The proof equips P with a generic strictly increasing rational-valued rank. Its stabilizer has ample generics, and generic pairs of ranks yield a comeager double coset. We show that the small index property passes from a closed subgroup with such a double coset to the ambient Polish group.
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BibTeX
@unpublished{Lupini202623452788,
title = {The small index property for the random poset},
author = {Martino Lupini},
year = {2026},
doi = {10.5281/zenodo.23452788},
note = {Zenodo deposit, latest of 2}
}