What question did this study set out to answer?

Investigate the relationships between compact Lie groups of homeomorphisms and their containment within orthogonal groups. The focus is on understanding the conditions under which these groups can be conjugate.

March 29, 2026Open Access

On Compact Abelian Lie Groups of Homeomorphisms of Rm

Key Points

Investigate the relationships between compact Lie groups of homeomorphisms and their containment within orthogonal groups. The focus is on understanding the conditions under which these groups can be conjugate.
Consider the action of the group on a specified structure
Analyze the conditions for elements of the group to centralize a subgroup
Explore properties of compact Lie groups in a higher dimensional context
Show that the naive conjecture of conjugacy to the orthogonal group is false in higher dimensions
Establish that G is contained in S if every element of G centralizes S
Provide partial answers regarding subgroup structures in the context of homeomorphisms

Abstract

Let G be a compact Lie group of homeomorphisms of R m .The Naive conjecture saying that G is conjugate to a subgroup of the orthogonal group O(m) is known to be false for higher dimension.In this paper we give a partial answer by considering the action of the groupfor 1 i q , and we show that G is contained in S if and only if every element of G centralizes S .

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