I asked some questions on a descending lemma in Lawrence-Venkatesh 4 days ago, but it has not received any answer. I understood (2) now but I'm still confused on (1).
I want to ask a new question here. In the same proof, they stated that $\Gamma=\pi_1^{\rm geo}(Y-\{y_0\})$ is a normal subgroup of $\tilde{\Gamma}^{\rm geo}(Y^2-\Delta,\bf{y})$, then they said that $\Gamma$ is the normal subgroup of $\tilde{\Gamma}=\pi_1^{\rm ari}((Y^2-\Delta)_K)$, I could not see why this holds.
For $H$ a normal subgroup of $G$, $G/H$ acts on the set of normal subgroups of $H$. Then $H$ is a fixed point of this action if and only if it is a normal subgroup of $G$.
The quotient in this case is the Galois group of the base field, so we want the subgroup to be invariant under the Galois group. But the Galois action sends the fundamental group of a subvariety to the fundamental group of a Galois conjugate subvariety so if the subvariety is defined over a base field then the subgroup is invariant under the Galois group.
