Let $G$ be a torsionfree kleinian group. Is there necessary an sufficient conditions on $G$ to be a knot group ?

It seems that there is some necessary conditions:

1. $H_{1}(BG) = \mathbb{Z}$
2. $H_{2}(BG) =H_{3}(BG) = 0$
3. $M= \mathbb{H}^3/G$ has a finite hyperbolic volume and $M$ has exactly one cusp.
4. $\mathbb{Z}^2$ is isomorphic to a subgroup of $G$.
5. (I am not sure about this one) for any non trivial element $x\in G$ , $ G/<x>$ is the trivial group, where $<x>$ is the smallest normal subgroup of $G$ generated by $x$.

Question1: are these conditions sufficient?

Question2: let $G$ and $H$ be knot groups of hyperbolic knots such that the hyperbolic volume of $\mathbb{H}^3/G$ and $\mathbb{H}^3/H$ are equal. Are $G$ and $H$ isomorphic?

Let $G$ be a torsionfree kleinian group. Is there necessary an sufficient conditions on $G$ to be a knot group ?

It seems that there is some necessary conditions:

1. $H_{1}(BG) = \mathbb{Z}$
2. $H_{2}(BG) =H_{3}(BG) = 0$
3. $M= \mathbb{H}^3/G$ has a finite hyperbolic volume and $M$ has exactly one cusp.
4. $\mathbb{Z}^2$ is isomorphic to a subgroup of $G$.
5. (I am not sure about this one) there exists an element $x\in G$ , $ G/<x>$ is the trivial group, where $<x>$ is the smallest normal subgroup of $G$ generated by $x$.

Question1: are these conditions sufficient?

Question2: let $G$ and $H$ be knot groups of hyperbolic knots such that the hyperbolic volume of $\mathbb{H}^3/G$ and $\mathbb{H}^3/H$ are equal. Are $G$ and $H$ isomorphic?

Let $G$ be a kleinian group. Is there necessary an sufficient conditions on $G$ to be a knot group ?

It seems that there is some necessary conditions:

1. $H_{1}(BG) = \mathbb{Z}$
2. $H_{2}(BG) =H_{3}(BG) = 0$
3. $M= \mathbb{H}^3/G$ has a finite hyperbolic volume and $M$ has exactly one cusp.
4. $\mathbb{Z}^2$ is isomorphic to a subgroup of $G$.
5. (I am not sure about this one) for any non trivial element $x\in G$ , $ G/<x>$ is the trivial group.

Question1: are these conditions sufficient?

Question2: let $G$ and $H$ be knot groups of hyperbolic knots such that the hyperbolic volume of $\mathbb{H}^3/G$ and $\mathbb{H}^3/H$ are equal. Are $G$ and $H$ isomorphic?

Let $G$ be a torsionfree kleinian group. Is there necessary an sufficient conditions on $G$ to be a knot group ?

It seems that there is some necessary conditions:

1. $H_{1}(BG) = \mathbb{Z}$
2. $H_{2}(BG) =H_{3}(BG) = 0$
3. $M= \mathbb{H}^3/G$ has a finite hyperbolic volume and $M$ has exactly one cusp.
4. $\mathbb{Z}^2$ is isomorphic to a subgroup of $G$.
5. (I am not sure about this one) for any non trivial element $x\in G$ , $ G/<x>$ is the trivial group, where $<x>$ is the smallest normal subgroup of $G$ generated by $x$.

Question1: are these conditions sufficient?

Question2: let $G$ and $H$ be knot groups of hyperbolic knots such that the hyperbolic volume of $\mathbb{H}^3/G$ and $\mathbb{H}^3/H$ are equal. Are $G$ and $H$ isomorphic?