Peter Jorgensen: SL₂-tilings, infinite triangulations, and continuous cluster categories

This is report on joint work with Christine Bessenrodt and Thorsten Holm. An \(SL_2\) tiling is an infinite grid of positive integers such that each adjacent 2x2-submatrix has determinant 1. These tilings were introduced by Assem, Reutenauer, and Smith for combinatorial purposes. We will show a bijection between \(SL_2\) tilings and certain infinite triangulations of the circle with four accumulation points. We will see how properties of the tilings are reflected in the triangulations. For instance, the entry 1 of a tiling always gives an arc of the corresponding triangulation, and 1 can occur infinitely often in a tiling. On the other hand, if a tiling has no entry equal to 1, then the minimal entry of the tiling is unique, and the minimal entry can be seen as a more complex pattern in the triangulation. The infinite triangulations also give rise to cluster tilting subcategories in a certain c luster category with infinite clusters related to the continuous cluster categories of Igusa and Todorov. The \(SL_2\) tilings can be viewed as the corresponding cluster characters.