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PCG Report Summary

Project ID: 336983
Funded under: FP7-IDEAS-ERC
Country: Spain

Periodic Report Summary 2 - PCG (The Elementary Theory of Partially Commutative Groups)

Hilbert’s 10th problem asks if there exists an algorithm to decide whether or not an equation with integer coefficients has an integer solution. This type of problems can be posed for arbitrary structures (rings, groups etc.) and in a more general setting from the viewpoint of Model Theory.

In the case of free groups, this kind of problem was posed by Tarski and the solution has uncovered deep connections between free groups and the geometry of trees and established a link between Model Theory and Geometry.

The framework of our project is an analogue of Tarski problem for a central class of groups, known as Right-Angled Artin Groups. In our work we introduce higher-dimensional generalisation of trees, which we call (real) cubings. We further relate the theory of Right-Angled Artin Groups with the geometry of real cubings.

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