Monod, p. 2 — G(ℤ) with rational breakpoints is exactly the set of its piecewise elements

A homeomorphism $f$ of the projective line $\mathbf P^1 = \mathbb R \cup \{\infty\}$ lies in GRat, the subgroup of Gpp generated by the homeomorphisms that are piecewise in $\mathrm{PSL}_2(\mathbb Z)$ with all breakpoints in $\mathbb Q \cup \{\infty\}$, exactly when $f$ itself is such a homeomorphism: $f \in$ Gpp and IsPiecewiseProjOn ⊥ ratPoints f. So the generated subgroup adds nothing: these elements already form a group.

GRat, Gpp, IsPiecewiseProjOn and ratPoints (the set $\mathbb Q \cup \{\infty\}$) are from the Monod definitions bundle, where ⊥ is the subring $\mathbb Z$ of $\mathbb R$.

Monod writes on p. 2: “The relation is as follows: if we modify the definition of $H(\mathbf{Z})$ by requiring that the breakpoints be rational, then all its elements are automatically $C^1$ and the resulting group is conjugated to $F$. The corresponding relation holds between $G(\mathbf{Z})$ and Thompson’s group $T$.” The bundle defines “$G(\mathbf{Z})$ with rational breakpoints” as a generated subgroup; this theorem says it is the set of rational-breakpoint elements, the group Monod's sentence speaks of, so that the published Thurston statements about GRat and HRat (conjugacy to Thompson's groups $T$ and $F$) are about Monod's groups. The proof follows the one for $G(A)$ (Monod.mem_G_iff_isPiecewiseProj): a Möbius map with integer entries sends $\mathbb Q \cup \{\infty\}$ to itself, so the breakpoints of a composite stay rational.