# Fundamental domains for proper Lie group actions on smooth manifolds

The setting: $$M$$ an arbitrary smooth manifold, $$G$$ a Lie group acting effectively and properly on $$M$$ by diffeomorphisms.

Motivation: when trying to figure out the homeomorphism type of the orbit space, one tries to find a "fundamental domain" $$D$$ that contains exactly one representative of each orbit except for a set of measure zero on which identifications may occur, in which case you have some orbits with more than one representative. You want the projection $$\pi:M \rightarrow M / G$$ to restrict to a quotient map on your fundamental domain so that you're sure $$D / G$$ is homeomorphic to $$M / G$$.

I have only seen good definitions for what is meant by "fundamental domain" in the case of properly discontinuous actions, and the conversation seems to focus on the free action of a discrete Lie group—see Misha's answer to Proper discontinuity and existence of a fundamental domain for example—, but I am interested in non-free actions of not-necessarily-discrete Lie groups, and researchers in my field take "fundamental domain" to be an undefined intuitive concept, so think of a "fundamental domain" in whatever way you see fit.

The question: can you think of any example of an action in this setting where you can restrict to a subset (no restrictions on the properties of the set, but the less pathological the better) $$D$$ that contains at least one representative of every orbit, but such that $$D / G$$ is not homeomorphic to $$M / G$$ or (equivalently?) $$\pi\rvert_D$$ fails to be a quotient map?

In other words, is it possible for the "fundamental domain" thinking to go wrong?

If not, can you prove a sense in which it always works?

• Answers can move, so it is better to link to a specific answer than to refer to the position of an answer. Since the question to which you linked had only one answer, I thought it was probably clear what was meant, and edited to change the link accordingly. Commented Nov 17, 2021 at 1:49
• Just saw your question. It seems that this paper may have an answer for you? I'm yet to read the paper, but in the introduction session, the author writes: "This is just a generalization of the classical concept of fundamental domain, usually formulated for G discrete. In this work we show that fundamental domains with nice properties exist for a large class of nondiscrete transformation groups: We assume that M is a complete, connected Riemannian manifold and that G is a closed subgroup of the group of isometries of M". Sorry if this isn't helpful! Commented Feb 21 at 14:37
• Sorry, but forgot to mention the paper: On the Existence of a Fundamental Domain for Riemannian Transformation Groups Author(s): Robert Hermann Source: Proceedings of the American Mathematical Society , Jun., 1962, Vol. 13, No. 3 (Jun., 1962), pp. 489-494 Published by: American Mathematical Society Stable URL: jstor.org/stable/2034968 Commented Feb 22 at 9:50