[Paper Review] Tits alternative for automorphism groups of compact Kaehler manifolds

Tien‐Cuong Dinh|arXiv (Cornell University)|Oct 13, 2012

Geometry and complex manifolds31 references3 citations

TL;DR

This paper establishes the Tits alternative for automorphism groups of compact Kähler manifolds, proving that any such group either contains a free non-abelian subgroup or has a finite-index solvable subgroup. The key result shows that the group of zero entropy automorphisms forms a normal subgroup, and the quotient is a free abelian group of rank at most $k - \kappa - 1$, where $k$ is the complex dimension and $\kappa$ depends on the Kodaira dimension. The proof relies on a mixed Hodge-Riemann theorem and dynamical degree analysis.

ABSTRACT

We survey some properties of the automorphism groups of compact Kaehler manifolds. In particular, we present recent results by Keum, Oguiso and Zhang on the structure of these groups from the Tits alternative point of view. Several other related results will be also discussed. This paper is written for the VIASM Annual Meeting 2012.

Motivation & Objective

• To establish the Tits alternative for automorphism groups of compact Kähler manifolds, confirming a conjecture by Keum-Oguiso-Zhang.
• To analyze the structure of automorphism groups under the assumption that they do not contain free non-abelian subgroups.
• To characterize the subgroup of zero entropy automorphisms and determine the rank of the quotient group of finite-index solvable subgroups.
• To relate the dynamical entropy of automorphisms to algebraic invariants such as dynamical degrees and cohomological actions.

Proposed method

• Utilizes a mixed version of the Hodge-Riemann theorem to analyze cohomological actions of automorphisms on Kähler manifolds.
• Applies the Perron-Frobenius theorem to invariant classes in the Kähler cone to construct invariant cohomology classes under group actions.
• Employs the concept of weak Hodge-Riemann (wHR) classes to ensure positivity and invariance under automorphism actions.
• Introduces a group morphism $\phi: G' \to \mathbb{R}^{k-\kappa-1}$ defined by logarithmic characters of the action on cohomology classes.
• Uses the inequality $\|\phi(g)\| \geq \frac{1}{2} \log d_{k-1}(g)$ to prove discreteness of the image, ensuring the quotient is a free abelian group.
• Relies on Yomdin-Gromov theory to equate topological entropy with dynamical degrees, linking dynamics to algebraic geometry.

Experimental results

Research questions

• RQ1 Does the automorphism group of a compact Kähler manifold satisfy the Tits alternative, i.e., either contain a free non-abelian subgroup or be virtually solvable?
• RQ2 What is the maximal rank of a free abelian subgroup of automorphisms with positive entropy?
• RQ3 How does the dynamical degree $d_{k-1}(g)$ relate to the cohomological action of $g^*$ on the Kähler cone?
• RQ4 Can the subgroup of zero entropy automorphisms be characterized as a normal subgroup within a finite-index solvable subgroup?
• RQ5 What is the precise bound on the rank of the quotient group $G'/N'$, and is it optimal?

Key findings

• The automorphism group $\mathrm{Aut}(X)$ of a compact Kähler manifold $X$ of dimension $k$ satisfies the Tits alternative: any subgroup without a free non-abelian subgroup is virtually solvable.
• For any such group $G$, there exists a finite-index subgroup $G'$ that is solvable, with the zero entropy elements forming a normal subgroup $N'$.
• The quotient $G'/N'$ is a free abelian group of rank at most $k - \kappa - 1$, where $\kappa = \max(\kappa_X, 0)$ if $\kappa_X < k$, and $\kappa = k-1$ otherwise.
• The rank bound is optimal, as demonstrated by examples from $\mathrm{SL}(k,\mathbb{Z})$ acting on complex tori, which yield free abelian subgroups of rank $k-1$ with positive entropy.
• The topological entropy of an automorphism $g$ is zero if and only if $\phi(g) = 0$, where $\phi$ is the morphism encoding the logarithmic eigenvalues of the action on cohomology.
• The image $\phi(G')$ is discrete in $\mathbb{R}^{k-\kappa-1}$, which ensures that $G'/N'$ is a free abelian group of finite rank.