Sofic profile and computability of Cremona groups

TL;DR

This paper proves Cremona groups are sofic, introduces a new quantitative measure called sofic profile, and demonstrates that certain subgroups have solvable word problems, advancing understanding of their algebraic properties.

Contribution

It introduces the concept of sofic profile for Cremona groups and establishes polynomial bounds, providing new insights into their structure and computability.

Findings

Cremona groups are sofic.

Sofic profile of birational transformation groups is polynomial of degree d.

Finitely generated subgroups of Cremona groups have solvable word problem.

Abstract

In this paper, we show that Cremona groups are sofic. We actually introduce a quantitative notion of soficity, called sofic profile, and show that the group of birational transformations of a d-dimensional variety has sofic profile at most polynomial of degree d. We also observe that finitely generated subgroups of the Cremona group have a solvable word problem. This provides examples of finitely generated groups with no embeddings into any Cremona group, answering a question of S. Cantat.