TITS ALTERNATIVE AND HIGHLY TRANSITIVE ACTIONS ON TORIC VARIETIES

Ivan Arzhantsev (1) , M Zaidenberg (2)

1 HSE - Vysšaja škola èkonomiki = National Research University Higher School of Economics [Moscow]
2 IF - Institut Fourier

Given a toric affine algebraic variety $X$ and a collection of one-parameter unipotent subgroups $U_1,\ldots,U_s$ of $\Aut(X)$ which are normalized by the torus acting on $X$, we show that the group $G$ generated by $U_1,\ldots,U_s$ verifies the Tits alternative, and, moreover, either is a unipotent algebraic group, or contains a nonabelian free subgroup. We deduce that if $G$ is $m$-transitive for any positive integer $m$, then $G$ contains a nonabelian free subgroup, and so, is of exponential growth.

Mots clés: one-parameter subgroup; group action; affine variety; toric variety; Demazure root; multiple transitivity

Domaines: Géométrie algébrique [math.AG]

Ivan Arzhantsev, M Zaidenberg. TITS ALTERNATIVE AND HIGHLY TRANSITIVE ACTIONS ON TORIC VARIETIES. 2020. ⟨hal-02494577v2⟩