On the biautomaticity of CAT(0) triangle-square groups

Following the research from the paper “Triangles, squares and geodesics” (arXiv:0910.5688) of Rena Levitt and Jon McCammond we investigate the properties of groups acting on CAT(0) triangle-square complexes, focusing mostly on biautomaticity of such groups. In particular we show two examples of nonpositively curved triangle-square complexes $X_1$ and $X_2$, such that their universal covers violate conjectures given in the aforementioned paper. This shows that the Gersten-Short geodesics cannot be used as a way of proving biautomaticity of groups acting on such complexes. Lastly we give a proof of biautomaticity of ${\pi }_1(X_1)$, however the biautomaticity of ${\pi }_1(X_2)$ remains unknown.