What's the meaning of the length of glide reflection? Problem 4 of this sheet states: > Let $\Gamma$ be a discontinuous, fixed point free group of glide reflections and translations. Let $g$ be a glide reflection of minimal length in $\Gamma$, and let $h$ be an element of minimal length in $\Gamma$ not in the direction of $g$. Prove that $g$ and $h$ must have perpendicular directions (e.g., by finding shorter elements when the directions of $g$ and $h$ are not perpendicular). This refers to a glide reflection with minimal length, but what's the definition of the length here, does that mean the length of the translation in the glide reflection?

I'd say yes, the length of the glide reflection is the translational component. If you apply a glide reflection twice you obtain a translation, and half it's distance is what I'd call the length or distance of the glide reflection.