Arkiv för Matematik

Volume 56 (2018)

Number 1

Infinite transitivity and special automorphisms

Pages: 1 – 14



Ivan Arzhantsev (Faculty of Computer Science, National Research University Higher School of Economics, Moscow, Russia)


It is known that if the special automorphism group $\mathrm{SAut}(X)$ of a quasiaffine variety $X$ of dimension at least $2$ acts transitively on $X$, then this action is infinitely transitive. In this paper we question whether this is the only possibility for the automorphism group $\mathbb{Aut}(X)$ to act infinitely transitively on $X$. We show that this is the case, provided $X$ admits a nontrivial $\mathbb{G}_a$ or $\mathbb{G}_m$-action. Moreover, $2$-transitivity of the automorphism group implies infinite transitivity.


quasiaffine variety, automorphism, transitivity, torus action, rigidity

2010 Mathematics Subject Classification

Primary 14J50, 14M17. Secondary 13A50, 14L30, 14R20.

The author’s research was supported by the grant RSF-DFG 16-41-01013.

Received 28 October 2016

Received revised 14 May 2017

Accepted 5 June 2017

Published 30 April 2018