# Triangle angle sums related to translation curves in $\SOL$ geometry

## TL;DR

This paper investigates the interior angle sums of translation triangles in $ ext{SOL}$ geometry, showing they can be greater than or equal to $ ext{pi}$, expanding understanding of geometric properties in Thurston geometries.

## Contribution

It provides the first analysis of translation triangle angle sums in $ ext{SOL}$ geometry, demonstrating they can exceed or equal $ ext{pi}$, using the projective model of $ ext{SOL}$ geometry.

## Key findings

- Angle sums can be larger than $ ext{pi}$ in $ ext{SOL}$ geometry.
- The study extends geometric analysis from $ ext{NIL}$ and $ ext{SLR}$ to $ ext{SOL}$.
- Uses the projective model of $ ext{SOL}$ geometry for analysis.

## Abstract

After having investigated the geodesic and translation triangles and their angle sums in $\NIL$ and $\SLR$ geometries we consider the analogous problem in $\SOL$ space that is one of the eight 3-dimensional Thurston geometries. We analyse the interior angle sums of translation triangles in $\SOL$ geometry and prove that it can be larger or equal than $\pi$. In our work we will use the projective model of $\SOL$ described by E. Moln\'ar in \cite{M97},