I'm not sure if I understand term "geodesic words" correctly. Need this to translate chapter about braid groups to polish, so if any of You know polish equivalent, please share it. Using wikipedia's definition for "Word (group theory)" i have:
For example, if x, y and z are elements of a group G, then xy, z−1xzz and y−1zxx−1yz−1 are words in the set {x, y, z}.
I know that geodesic line is the shortest line between two given points.
Adding both informations i think that "geodesic word" means the shortest word, one in which all possible cancellation was done.
Example: $xz$ - geodesic, $xyy^{-1}z$ - not geodesic. Am I right? If not, please help me out.
If the group $G$ is generated by the set $X$, then a word $w = a_1a_2 \cdots a_n$ with each $a_i \in X \cup X^{-1}$ is a geodesic word (with respect to the group $G$) if there is no shorter word that represents the same group element as $w$.
So, for example, in the braid group $\langle x,y \mid xyx=yxy \rangle$, the words $xyx$ and $yxy$ (which represent the same group element) are both geodesics, but $x^{-1}y^{-1}xy$ is not, because $yx^{-1}$ is shorter, and represents the same group element.
Such words are called geodesics because they label geodesic paths in the Cayley graph of $G$ w.r.t. $X$.