Dark matter's effect in 2+1 GR? In the appendix to The Planiverse, it is acknowledged that GR in 2 dimensional space implies no gravitational forces between separated masses--only in the interior of extended massive bodies. The author then speculates that perhaps the natives of the Planiverse will someday discover dark matter as an explanation for why their disc-planet in fact has gravity following a $\frac{1}{r}$ law.
But, is that how it actually works? If 2d dark matter were concentrated in a localized halo around a 2d planet, I would expect it to produce a linearly-rising gravitational force everywhere within the halo (or approximately so, modulo density variations in the matter field), just like the field inside our own planet. So that leaves the case of dark matter being distributed uniformly throughout the entire 2d cosmos, so that the dark matter itself produces no local accelerations; would the simple presence of dark matter, such that every point in space is indeed in the interior of a massive body, result in gravity appearing to behave normally for local concentrations of mass like planets, or would it simply be locally unnoticeable?
 A: The case of lower-dimensional general relativity is an interesting one because due to the symmetries of the Riemann tensor, things simplify quite a lot. In three dimensions in particular, the Riemann tensor has $6$ independent components, which is also the number of components of the Ricci tensor. It can be shown that, in 3 dimensions, the Riemann tensor is simply
$$R_{abcd} = R_{ac}g_{bd} - R_{ad}g_{bc} + R_{bd} g_{ac} - R_{bc} g_{ad} - \frac{1}{2} R (g_{ac}g_{bd} - g_{ad}g_{bd})$$
or equivalently, that the Weyl tensor vanishes. Very roughly, the Weyl tensor corresponds in general relativity to the notion that gravity can propagate in a vacuum, since the Riemann tensor can always be decomposed in three terms, one depending on the Ricci tensor, one depending on the Ricci scalar, and one depending on the Weyl tensor. And as we well know, in GR, we have that, if $T_{ab} = 0$, then $R_{ab} = 0$. So that, in $2+1$ dimensions, we have
$$R_{abcd} = 0$$
This doesn't necessarily correspond to Minkowski space (and indeed spacetime isn't Minkowski space in $2+1$ dimensions for the Schwarzschild spacetime, as it is not topologically trivial), but it does mean that there are no tidal forces and no gravitational attraction : the metric is just locally Minkowski.
Adding a cosmological constant, or just a pervasive matter field, does indeed help to get some more typical gravitational forces, as can be seen in the BTZ black hole solution, for instance.
