Geodesic
THE ${\it\alpha}$ -INVARIANT AND THOMPSON’S CONJECTURE
Forum of Mathematics, Pi, Tome 4 (2016)

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In 1981, Thompson proved that, if $n\geqslant 1$ is any integer and $G$ is any finite subgroup of $\text{GL}_{n}(\mathbb{C})$ , then $G$ has a semi-invariant of degree at most $4n^{2}$ . He conjectured that, in fact, there is a universal constant $C$ such that for any $n\in \mathbb{N}$ and any finite subgroup $G\text{GL}_{n}(\mathbb{C})$ , $G$ has a semi-invariant of degree at most $Cn$ . This conjecture would imply that the ${\it\alpha}$ -invariant ${\it\alpha}_{G}(\mathbb{P}^{n-1})$ , as introduced by Tian in 1987, is at most $C$ . We prove Thompson’s conjecture in this paper. 
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     author = {PHAM HUU TIEP},
     title = {THE ${\it\alpha}$ {-INVARIANT} {AND} {THOMPSON{\textquoteright}S} {CONJECTURE}},
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PHAM HUU TIEP. THE ${\it\alpha}$ -INVARIANT AND THOMPSON’S CONJECTURE. Forum of Mathematics, Pi, Tome 4 (2016). doi: 10.1017/fmp.2016.3

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