Euler class of taut foliations and Dehn filling

In this article, we study the Euler class of taut foliations on the Dehn fillings of a $\mathbb{Q}$-homology solid torus. We give a necessary and sufficient condition for the Euler class of a foliation transverse to the core of the filling solid torus to vanish. We apply this condition to taut foliations on Dehn fillings of hyperbolic fibered manifolds and obtain many new left-orderable Dehn filling slopes on these manifolds. For instance, we show that when $X$ is the exterior of a pretzel knot $P(-2,3,2r+1)$, $r\geq 3$, $π_1(X(α_n))$ is left-orderable for a sequence of positive slopes $α_n$ with $α_0 =2g-2$ and $α_n\to 2g-1$. Lastly, we prove that given any $\mathbb{Q}$-homology solid torus, the set of slopes for which the corresponding Dehn fillings admit a taut foliation transverse to the core with zero Euler class is nowhere dense in $\mathbb{R}\cup \{\frac{1}{0}\}$.

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