is equivalent to the fact that the quotient $G/X$ is compact (sometimes called $G$-compactness or co-compactness), which is also equivalent to $X$ being compact.

Let $G$ be a discrete group. Recall that given a $G$-CW complex $X$, the $G$-action on $X$ is proper exactly when all point stabilizers are finite. This is equivalent to slice conditions due to the slice theorem [@palais], [@antonyan].

\[defgeomk\]\[Geometric Twisted Equivariant K-Homology\]

Let $G$ be a discrete group and let $X$ be a proper $G$-CW complex, let $P$ be a projective, unitary stable $G$-equivariant bundle.

A geometric cycle is a triple $(M,f,\sigma)$ consisting of

-   A $G$-compact, proper orientable, ${\operatorname{Spin}}(c)$ manifold without boundary $M$.

-   A $G$-equivariant map $f: M\to X$.

-   A homotopy class of a $G$-invariant section $\sigma:  M\to {\ensuremath{{\mathrm{Fred}}}}(-f^*(P))$ representing an element on the twisted equivariant $K$-theory group $$K_G^0(M, -f^*(P)).$$

We will impose some conditions on geometric cycles in order to obtain geometric Twisted Equivariant $K$-Homology.

1.  Isomorphism. The cocycle $(M,f, \sigma)$ is isomorphic to $(M^{'},f^{'}, \sigma^{'})$ if there exists a $G$-equivariant, ${\operatorname{Spin}}(c)$-structure preserving diffeomorphism $\psi: M\to  M^{'}$, which makes the following diagram commutative:

    $$\xymatrix{M   \ar[rr]^-{\psi} \ar[rd]_-{f} & & M^{'}\ar[dl]^-{f^{'}}\\ & X &   
     }$$

2.  Bordism. The cycle $(M_0, f_0,\sigma_0 )$is bordant 