Search results for " hyperbolic curve"

MR 3079286 Reviewed Hoshi Y. On a problem of Matsumoto and Tamagawa concerning monodromic fullness of hyperbolic curves: genus zero case. Tohoku Math…

2014

Let \emph{Primes} be the set of all prime numbers, $k$ be a finite extension of the field of rational numbers and $\bar{k}$ be an algebraic closure of $k$. Let $(g, r)$ be a pair of nonnegative integers such that $2g - 2 + r > 0$ and $X$ be a hyperbolic curve of type $(g, r)$ over $k$. The author observes that, for each $l \in \emph{Primes}$, there are two natural outer representations on $\pi^{\{ l\}}_{1} (X \otimes_{k} \bar{k})$: $$\rho_{X / k} ^{\{ l\}}: G_{k} := Gal(\bar{k} / k) \rightarrow Out (\pi^{\{ l\}}_{1} (X \otimes_{k} \bar{k}))$$ and $$ \rho_{g, [r]} ^{\{ l\}}: \pi_{1}(M_{g, [r]}) \rightarrow Out (\pi^{\{ l\}}_{1} (X \otimes_{k} \bar{k})),$$ where $\pi^{\{ l\}}_{1} (X \otimes_{…