Automaticity of non-positively curved $k$-fold triangle groups

TL;DR

This work proves that non-positively curved $k$-fold triangle groups have finitely many cone types, which implies a regular language of all geodesics. It then shows that the language of lexicographically first geodesics is regular and satisfies the fellow traveller property, establishing automaticity for these groups. The key approach combines disc diagrams, angle structures, and a precise relation between combinatorial geodesics and CAT$(0)$ geodesics to transfer local geometric data into a finite-state, computable framework. These results extend automaticity insights beyond cubical groups and enable explicit computations for this class of groups, including potential word problem algorithms and structural analyses.

Abstract

We show that non-positively curved $k$-fold triangle groups have finitely many cone types, and hence a regular language of all geodesics. Further, we prove that the language of lexicographically first geodesics is both regular and satisfies the fellow traveller property, giving an automatic structure for this family of groups.

Figures (10)

• Figure 1: A $k$-fold triangle group with vertex groups $V_1$,$V_2$ and $V_3$
• Figure 2: A part of the complex $X$ for a 3-fold triangle group
• Figure 3: Comparing the directions of $\gamma$ and $g_{\gamma}$
• Figure 4: Retracting over a hexagon with two positively curved vertices and no $g$-adjacent triangles.
• Figure 5: Retriangulating to get a $g$-adjacent triangle which shares an edge with the polygon.

Theorems & Definitions (37)

• Theorem 1.1
• Lemma 1.2
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Stallings, Stallings91
• Remark 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.7
• Remark 2.8