Algebraic and geometric properties of homeomorphism groups of ordinals

2024/12/22 by Bhat, Megha, Ruru Chen, Adityo Mamun +8
Business, Management and Accounting · Decision Sciences · #20F38 #54G12 #57S05 #FOS: Mathematics #Fuzzy and Soft Set Theory #General Topology (math.GN) #Geometric Topology (math.GT) #Group Theory (math.GR) #Optics and Image Analysis

paper · pdf · doi:10.48550/arxiv.2412.17103

openalex publication_date 2024/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the homeomorphism groups of ordinals equipped with their order topology, focusing on successor ordinals whose limit capacity is also a successor. This is a rich family of groups that has connections to both permutation groups and homeomorphism groups of manifolds. For ordinals of Cantor--Bendixson degree one, we prove that the homeomorphism group is strongly distorted and uniformly perfect, and we classify its normal generators. As a corollary, we recover and provide a new proof of the classical result that the subgroup of finite permutations in the symmetric group on a countably infinite set is the maximal proper normal subgroup. For ordinals of higher Cantor--Bendixson degree, we establish a semi-direct product decomposition of the (pure) homeomorphism group. When the limit capacity is one, we further compute the abelianizations and determine normal generating sets of minimal cardinality for these groups.

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