Note that since you ask for simply-connected groups, you should use the universal cover $\widetilde{PSL}(2, R)$ of $PSL(2, R)$ rather than $SL(2, R)$.

The question of what compact $3$-manifolds are quotients of a simply-connected Lie group by a discrete subgroup was answered by F. Raymond and A. Vasquez in "3-Manifolds whose universal coverings are Lie groups" published in Topology and its Applications 12 (1981) 161-179. (In fact they implicitly answered the more general question in which $G/\Gamma$ has finite volume.) Almost all of the resulting three manifolds are Seifert manifolds and $\Gamma$ is described in terms of the Seifert invariants of the quotient. The few cases not describable this way are quotients of solvable Lie groups. A Lie group that admits a finite volume quotient by a discrete group is unimodular (this is proved in Milnor's paper on left-invariant metrics) and the unimodular three-dimensional Lie groups were classified by Bianchi (L. Bianchi, Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti., Memorie della Società Italiana delle Scienze (detta dei XL) 11 (1898), 267–352.). In addition to the abelian case, nilpotent case (Heisenberg), and the semisimple cases $SU(2)$ and $\widetilde{PSL}(2, R)$, G can also be the (universal cover of the) groups of Euclidean motions of the plane or the group of affine motions preserving a split signature bilinear form on the plane (these are solvable but not nilpotent).

$\DeclareMathOperator\PSL{PSL}$Note that since you ask for simply-connected groups, you should use the universal cover $\widetilde\PSL(2, R)$ of $\PSL(2, R)$ rather than $\operatorname{SL}(2, R)$.

The question of what compact $3$-manifolds are quotients of a simply-connected Lie group by a discrete subgroup was answered by F. Raymond and A. Vasquez in "3-Manifolds whose universal coverings are Lie groups" published in Topology and its Applications 12 (1981) 161-179. (In fact they implicitly answered the more general question in which $G/\Gamma$ has finite volume.) Almost all of the resulting three manifolds are Seifert manifolds and $\Gamma$ is described in terms of the Seifert invariants of the quotient. The few cases not describable this way are quotients of solvable Lie groups. A Lie group that admits a finite volume quotient by a discrete group is unimodular (this is proved in Milnor's paper Curvatures of left invariant metrics on Lie groups) and the unimodular three-dimensional Lie groups were classified by Bianchi (L. Bianchi, Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti, Memorie della Società Italiana delle Scienze (detta dei XL) 11 (1898), 267–352). In addition to the abelian case, nilpotent case (Heisenberg), and the semisimple cases $\operatorname{SU}(2)$ and $\widetilde\PSL(2, R)$, $G$ can also be the (universal cover of the) groups of Euclidean motions of the plane or the group of affine motions preserving a split signature bilinear form on the plane (these are solvable but not nilpotent).

Note that since you ask for simply-connected groups, you should use the universal cover $\widetilde{PSL}(2, R)$ of $PSL(2, R)$ rather than $SL(2, R)$.

The question of what compact $3$-manifolds are quotients of a simply-connected Lie group by a discrete subgroup was answered by F. Raymond and A. Vasquez in "3-Manifolds whose universal coverings are Lie groups" published in Topology and its Applications 12 (1981) 161-179. (In fact they implicitly answered the more general question in which $G/\Gamma$ has finite volume.) Almost all of the resulting three manifolds are Seifert manifolds and $\Gamma$ is described in terms of the Seifert invariants of the quotient. The few cases not describable this way are quotients of solvable Lie groups. G must be unimodular and the unimodular three-dimensional Lie groups were classified by Bianchi (L. Bianchi, Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti., Memorie della Società Italiana delle Scienze (detta dei XL) 11 (1898), 267–352.). In addition to the abelian case, nilpotent case (Heisenberg), and the semisimple cases $SU(2)$ and $\widetilde{PSL}(2, R)$, G can also be the (universal cover of the) groups of Euclidean motions of the plane or the group of affine motions preserving a split signature bilinear form on the plane (these are solvable but not nilpotent).

Note that since you ask for simply-connected groups, you should use the universal cover $\widetilde{PSL}(2, R)$ of $PSL(2, R)$ rather than $SL(2, R)$.

The question of what compact $3$-manifolds are quotients of a simply-connected Lie group by a discrete subgroup was answered by F. Raymond and A. Vasquez in "3-Manifolds whose universal coverings are Lie groups" published in Topology and its Applications 12 (1981) 161-179. (In fact they implicitly answered the more general question in which $G/\Gamma$ has finite volume.) Almost all of the resulting three manifolds are Seifert manifolds and $\Gamma$ is described in terms of the Seifert invariants of the quotient. The few cases not describable this way are quotients of solvable Lie groups. A Lie group that admits a finite volume quotient by a discrete group is unimodular (this is proved in Milnor's paper on left-invariant metrics) and the unimodular three-dimensional Lie groups were classified by Bianchi (L. Bianchi, Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti., Memorie della Società Italiana delle Scienze (detta dei XL) 11 (1898), 267–352.). In addition to the abelian case, nilpotent case (Heisenberg), and the semisimple cases $SU(2)$ and $\widetilde{PSL}(2, R)$, G can also be the (universal cover of the) groups of Euclidean motions of the plane or the group of affine motions preserving a split signature bilinear form on the plane (these are solvable but not nilpotent).