Note that if a group $G$ acts on a space $X$ then $G$ also acts on the fundamental groupoid $\pi_1(X)$.

The study of the fundamental groupoid of an orbit space is developed in Chapter 11 of Topology and Groupoids, (T&G). An older version of this Chapter is available at arXiv:math/0212271.

The main result is to link the fundamental groupoid of an orbit space to the notion of orbit groupoid. The analysis of orbit groupoids uses fibrations of groupoids, which are also applied in Chapter 7 of T&G to yield operations of fundamental groupoids and exact sequences.

Here is a main result on the orbit groupoid $\Gamma //G$ of an action of a group $G$ on a groupoid $\Gamma$. (It comes from J. Taylor, "Quotients of a groupoid by the action of a group", Math. Proc. Camb. Phil. Soc 103 (1988) 239-249.)

11.5.2 The orbit morphism $p : \Gamma \to \Gamma // G$ is a fibration whose kernel is generated as a subgroupoid of $\Gamma$ by all elements of the form $\gamma - g\cdot \gamma$ where $g$ stabilises the initial point of $\gamma$. Furthermore,

(a) if $G$ acts freely on $\Gamma$, by which we mean no non-identity element of $G$ fixes an object of $\Gamma$, then $p$ is a covering morphism;

(b) if $\Gamma$ is connected and $G$ is generated by those of its elements which fix some object of $\Gamma$, then $p$ is a quotient morphism; in particular, $p$ is a quotient morphism if the action of $G$ on $Ob(\Gamma)$ has a fixed point;

(c) if $\Gamma$ is a tree groupoid, then each object group of $\Gamma // G$ is isomorphic to the factor group of $G$ by the (normal) subgroup of $G$ generated by elements which have fixed points.

You can also find out more on applications of fibrations of groupoids from arxiv:1207.6404.

Note that if a group $G$ acts on a space $X$ then $G$ also acts on the fundamental groupoid $\pi_1(X)$.

The study of the fundamental groupoid of an orbit space is developed in Chapter 11 of Topology and Groupoids, (T&G). An older version of this Chapter is available at arXiv:math/0212271.

The main result is to link the fundamental groupoid of an orbit space to the notion of orbit groupoid. The analysis of orbit groupoids uses fibrations of groupoids, which are also applied in Chapter 7 of T&G to yield operations of fundamental groupoids and exact sequences.

Here is a main result on the orbit groupoid $\Gamma //G$ of an action of a group $G$ on a groupoid $\Gamma$. (It comes from J. Taylor, "Quotients of a groupoid by the action of a group", Math. Proc. Camb. Phil. Soc 103 (1988) 239-249.)

11.5.2 The orbit morphism $p : \Gamma \to \Gamma // G$ is a fibration whose kernel is generated as a subgroupoid of $\Gamma$ by all elements of the form $\gamma - g\cdot \gamma$ where $g$ stabilises the initial point of $\gamma$. Furthermore,

(a) if $G$ acts freely on $\Gamma$, by which we mean no non-identity element of $G$ fixes an object of $\Gamma$, then $p$ is a covering morphism;

(b) if $\Gamma$ is connected and $G$ is generated by those of its elements which fix some object of $\Gamma$, then $p$ is a quotient morphism; in particular, $p$ is a quotient morphism if the action of $G$ on $Ob(\Gamma)$ has a fixed point;

(c) if $\Gamma$ is a tree groupoid, then each object group of $\Gamma // G$ is isomorphic to the factor group of $G$ by the (normal) subgroup of $G$ generated by elements which have fixed points.

You can also find out more on applications of fibrations of groupoids from arxiv:1207.6404.

Note that if a group $G$ acts on a space $X$ then $G$ also acts on the fundamental groupoid $\pi_1(X)$.

The study of the fundamental groupoid of an orbit space is developed in Chapter 11 of Topology and Groupoids, (T&G). An older version of this Chapter is available at arXiv:math/0212271.

The main result is to link the fundamental groupoid of an orbit space to the notion of orbit groupoid. The analysis of orbit groupoids uses fibrations of groupoids, which are also applied in Chapter 7 of T&G to yield operations of fundamental groupoids and exact sequences.

Here is a main result on the orbit groupoid $\Gamma //G$ of an action of a group $G$ on a groupoid $\Gamma$.

11.5.2 The orbit morphism $p : \Gamma \to \Gamma // G$ is a fibration whose kernel is generated as a subgroupoid of $\Gamma$ by all elements of the form $\gamma - g\cdot \gamma$ where $g$ stabilises the initial point of $\gamma$. Furthermore,

(a) if $G$ acts freely on $\Gamma$, by which we mean no non-identity element of $G$ fixes an object of $\Gamma$, then $p$ is a covering morphism;

(b) if $\Gamma$ is connected and $G$ is generated by those of its elements which fix some object of $\Gamma$, then $p$ is a quotient morphism; in particular, $p$ is a quotient morphism if the action of $G$ on $Ob(\Gamma)$ has a fixed point;

(c) if $\Gamma$ is a tree groupoid, then each object group of $\Gamma // G$ is isomorphic to the factor group of $G$ by the (normal) subgroup of $G$ generated by elements which have fixed points.

You can also find out more on applications of fibrations of groupoids from arxiv:1207.6404.

Note that if a group $G$ acts on a space $X$ then $G$ also acts on the fundamental groupoid $\pi_1(X)$.

The study of the fundamental groupoid of an orbit space is developed in Chapter 11 of Topology and Groupoids, (T&G). An older version of this Chapter is available at arXiv:math/0212271.

The main result is to link the fundamental groupoid of an orbit space to the notion of orbit groupoid. The analysis of orbit groupoids uses fibrations of groupoids, which are also applied in Chapter 7 of T&G to yield operations of fundamental groupoids and exact sequences.

Here is a main result on the orbit groupoid $\Gamma //G$ of an action of a group $G$ on a groupoid $\Gamma$. (It comes from J. Taylor, "Quotients of a groupoid by the action of a group", Math. Proc. Camb. Phil. Soc 103 (1988) 239-249.)

11.5.2 The orbit morphism $p : \Gamma \to \Gamma // G$ is a fibration whose kernel is generated as a subgroupoid of $\Gamma$ by all elements of the form $\gamma - g\cdot \gamma$ where $g$ stabilises the initial point of $\gamma$. Furthermore,

(a) if $G$ acts freely on $\Gamma$, by which we mean no non-identity element of $G$ fixes an object of $\Gamma$, then $p$ is a covering morphism;

(b) if $\Gamma$ is connected and $G$ is generated by those of its elements which fix some object of $\Gamma$, then $p$ is a quotient morphism; in particular, $p$ is a quotient morphism if the action of $G$ on $Ob(\Gamma)$ has a fixed point;

(c) if $\Gamma$ is a tree groupoid, then each object group of $\Gamma // G$ is isomorphic to the factor group of $G$ by the (normal) subgroup of $G$ generated by elements which have fixed points.

You can also find out more on applications of fibrations of groupoids from arxiv:1207.6404.