**Definition** : A group is **CAT(0)** if it acts properly, cocompactly and isometrically on a CAT(0) space.

**Examples** : see this blog.

**Remark** : A CAT(0) space is uniquely geodesic, but the converse is false (see here).

**Remark** : Every finitely-presented group is **geodesic**, i.e. acts properly, cocompactly and isometrically on a geodesic space (for example its Cayley graph related to a finite generating set).

**Definition** : A group is **uniquely geodesic** if it acts properly, cocompactly and isometrically on a uniquely geodesic space.

Question: Are torsion-free finitely presented groups, uniquely geodesic ? (see the next remark)

**Remark** : we add the assumption "with a proper action on a contractible CW complex so that the action is cocompact on each skeleton" (*see Misha's comment*). In addition, if groups with $\infty$-dimensional classifying space (as Thompson groups), give obvious counter-examples, we add the assumption "with a finite-dimensional classifying space". Perhaps the "cocompact" assumption is not relevant in the $\infty$-dimensional case, and should be replaced by something else.

**Remark** : The Baumslag-Solitar groups $BS(m,n)$ are torsion-free, finitely generated (with $2$-dim. classifying space) but not CAT(0) for $m \ne n$. Perhaps they are counter-examples, I don't know...

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continuously uniquely geodesic groups:

**Definition** : A continuously uniquely geodesic space is a uniquely geodesic space whose geodesics vary continuously with endpoints.

**Remark** : A continuously uniquely geodesic space is contractible.

**Remark** : A proper uniquely geodesic space is continuously uniquely geodesic, but a complete uniquely geodesic space is not necessarily continuously uniquely geodesic: see here page 38 (3.13 and 3.14)

**Definition** : A group is **continuously uniquely geodesic** if it acts properly, cocompactly and isometrically on a continuously uniquely geodesic space.

**Question** : Is a uniquely geodesic (finitely presented) group also continuously uniquely geodesic ?

**Remark**: In my opinion, the continuously uniquely geodesic (finitely presented) groups are very convenient with operator algebras conjectures as the Baum-Connes conjecture.