Definition question of convex orbit of finite group action

Question

Assume that a finite group or discrete group $G$ acts on a manifold $M$.

Here what does it mean that orbit $G\cdot x$ is convex ?

Thank you in advance.

Answer 1

It has been clarified that the question arose in the context of the paper Structure of fundamental groups of manifolds with Ricci curvature bounded below by Vitali Kapovitch and Burkhard Wilking. In the context of the paper, it seems that the authors use convexity in the following sense:

A metric space $X$ is convex if for any two points $x,y\in X$ and any $\lambda\in [0,1]$ there exists a point $z\in X$ such that $d(x,z)=\lambda\, d(x,y)$ and $d(y,z)=(1-\lambda)\, d(x,y)$.

A subset of a metric space is convex if it's a convex metric space with the restriction metric. This property is sometimes called metric convexity, but be aware that this terminology is inconsistent in the literature.

In the proof of Gap Lemma in the paper the authors disprove the convexity of a particular set (an orbit) by showing that it contains to points $z$ with $d(z,o)<0.51$ and $d(z,go)<0.51$, where $o$ and $go$ are two elements of the set with distance $1$ between them.

Then they arrive at contradiction by showing that orbits are actually of the form $\{v\}\times Z$ and therefore convex; I could not follow this claim because I can't see what $Z$ is on page 11.

In any event, the group whose orbits are considered here is obtained from discrete group by a limiting process; it need not be discrete itself.