Embeddings of SL(2,Z) into the Cremona group

TL;DR

This paper investigates the embeddings of SL(2,Z) into the Cremona group, revealing infinitely many non-conjugate embeddings that preserve element types and constructing explicit examples with specific geometric properties.

Contribution

It demonstrates the existence of infinitely many non-conjugate embeddings of SL(2,Z) into the Cremona group, preserving element types and providing explicit geometric examples.

Findings

Existence of infinitely many non-conjugate embeddings preserving element types

Construction of a specific automorphism group of a blown-up surface isomorphic to SL(2,Z)

Identification of embeddings that preserve elliptic, parabolic, and hyperbolic elements

Abstract

Geometric and dynamic properties of embeddings of SL(2,Z) into the Cremona group are studied. Infinitely many non-conjugate embeddings which preserve the type (i.e. which send elliptic, parabolic and hyperbolic elements onto elements of the same type) are provided. The existence of infinitely many non-conjugate elliptic, parabolic and hyperbolic embeddings is also shown. In particular, a group G of automorphisms of a smooth surface S obtained by blowing-up 10 points of the complex projective plane is given. The group G is isomorphic to SL(2,Z), preserves an elliptic curve and all its elements of infinite order are hyperbolic.