TITLE: Gluing hexagons to get a locally CAT(0) space
QUESTION [21 upvotes]: I believe that there are four ways to glue (all) the edges of a regular Euclidean hexagon to get a locally CAT(0) space:

The first two give the torus and the Klein bottle, respectively. What are the last two? In particular, do their fundamental groups have another name? Do they have the same fundamental group?

EDIT: HJRW points out that I missed the several ways to glue a hexagon to get a non-orientable surface of Euler characteristic -1. I count 8 possibilities up to symmetries.

REPLY [3 votes]: The fourth example was studied by Brady and Crisp in their CMH paper CAT(0) and CAT(-1) dimensions of torsion-free hyperbolic groups, so it would be reasonable to call its fundamental group the "Brady--Crisp group".  (Brady and Crisp also note that it belongs to a family studied by Haglund and Ballmann--Brin.)
They study a one-parameter family of CAT(0) metrics on the complex, and prove the very nice fact that any CAT(-1) model for this group has to have dimension at least 3.  (And they exhibit a 3-dimensional CAT(-1) model.)