I am looking at the variety of representations of the fundamental group of a hyperbolic 3-manifold into $\mathrm{SL}(n,\mathbb{C})$:
$$\mathrm{Hom}(\pi\_1(M), \mathrm{SL}(n,{\mathbb C}))$$


It is known that volume and Chern-Simons invariant of representations are constant on connected components of the representation variety. In particular, the representation variety has at least 3 components because complex conjugation of the representation changes the sign of the volume (and the trivial rep has zero volume).


What else is known about the number of connected components? Are there some lower bounds, perhaps coming from other invariants?



Representation varieties of 3-manifold groups in $\mathrm{SL}(n,\mathbb{C})$
These are difficult questions and very little in general is known about this. Mostly, what's known is ad hoc results for specific classes of manifolds (say, take some surgeries on 2-bridge knots...). You can find some references in answers to [this](https://mathoverflow.net/questions/88195/what-is-explicitly-known-about-the-sln-c-character-varieties-of-3-manifolds?rq=1) related question. 


Here is one general construction of connected components which, in general, cannot be reduced to the volume or the CS invariant. 


Let $M$ be a closed oriented hyperbolic 3-manifold (I assume this is what you are asking about). Then the holonomy representation $\rho: \pi\to SL(2,C)$ of its hyperbolic structure is locally rigid in $Hom(\pi, SL(2,C))$ (and the corresponding component can be detected by maximality of the volume). WLOG, we can assume that the field $K$ generated by the matrix coefficients of $\rho(\gamma), \gamma\in \pi$, is an algebraic number field (you can always achieve this by conjugation).


Then, you get other locally rigid representations $\rho^\sigma: \pi\to SL(2,C)$ by applying to $\rho$ elements $\sigma$ of the Galois group $Gal(K,Q)$. The reason is that the group $H^1(\pi, Ad\rho^\sigma)\cong H^1(\pi, Ad\rho)$ vanishes. The resulting representations $\rho^\sigma$ are (obviously) faithful but never discrete. Some of these representations will land in a compact subgroup of $SL(2,C)$ (this will always happen if $M$ is arithmetic), some will land in a subgroup conjugate to $SL(2,R)$ (and some will in neither of these groups). Such non-Zariski dense representations will have zero volume and CS invariant. Thus, these invariant cannot distinguish components of such representations, from say, the trivial one. 


