Suppose that a finite group $G$ acts on a proper (closed balls are compact) uniquely geodesic space $X$ by isometries. Then is there a point fixed by all of $G$?

This is true if $X$ is non-positively curved in the sense of Busemann (in particular for $CAT(0)$ spaces). Given any $y\in X$, the function $$ f(x) = \sum_{g\in G} d(x,gy)^2 $$ is proper and strictly convex, hence achieves its minimum at a unique point. The latter is fixed by $G$ since $f$ is invariant under $G$. (This is taken from Busemann's "The geometry of geodesics")

The result is also true when $X$ is Teichmüller space equipped with the Teichmüller metric. By a theorem of Royden-Earle-Kra, any isometry is induced by an element of the extended mapping class group. Moreover, any finite subgroup of the extended mapping class group has a fixed point in Teichmüller space. This is known as the Nielsen realization problem, solved by Kerckhoff.

Since the proofs for Teichmüller space are very specific to the geometry of surfaces, I suspect that the answer might be ''no'' in general. However, I don't know many examples of uniquely geodesic spaces besides the above. One way to construct new examples is to take $\ell^p$ products of uniquely geodesic spaces for $1<p<\infty$, but I don't think this would yield a counterexample.

Thus a subquestion is: What are good examples of uniquely geodesic spaces which are not Busemann spaces, Teichmüller spaces, or products thereof?

Edit: someone asked this question in the case G is cyclic here.

Suppose that a finite group $G$ acts on a proper (closed balls are compact) uniquely geodesic space $X$ by isometries. Then is there a point fixed by all of $G$?

This is true if $X$ is non-positively curved in the sense of Busemann (in particular for $CAT(0)$ spaces). Given any $y\in X$, the function $$ f(x) = \sum_{g\in G} d(x,gy)^2 $$ is proper and strictly convex, hence achieves its minimum at a unique point. The latter is fixed by $G$ since $f$ is invariant under $G$. (This is taken from Busemann's "The geometry of geodesics")

The result is also true when $X$ is Teichmüller space equipped with the Teichmüller metric. By a theorem of Royden-Earle-Kra, any isometry is induced by an element of the extended mapping class group. Moreover, any finite subgroup of the extended mapping class group has a fixed point in Teichmüller space. This is known as the Nielsen realization problem, solved by Kerckhoff.

Since the proofs for Teichmüller space are very specific to the geometry of surfaces, I suspect that the answer might be ''no'' in general. However, I don't know many examples of uniquely geodesic spaces besides the above. One way to construct new examples is to take $\ell^p$ products of uniquely geodesic spaces for $1<p<\infty$, but I don't think this would yield a counterexample.

Thus a subquestion is: What are good examples of uniquely geodesic spaces which are not Busemann spaces, Teichmüller spaces, or products thereof?

Edit: someone asked this question in the case G is cyclic here.

Suppose that a finite group $G$ acts on a proper (closed balls are compact) uniquely geodesic space $X$ by isometries. Then is there a point fixed by all of $G$?

This is true if $X$ is non-positively curved in the sense of Busemann (in particular for $CAT(0)$ spaces). Given any $y\in X$, the function $$ f(x) = \sum_{g\in G} d(x,gy)^2 $$ is proper and strictly convex, hence achieves its minimum at a unique point. The latter is fixed by $G$ since $f$ is invariant under $G$.

The result is also true when $X$ is Teichmüller space equipped with the Teichmüller metric. By a theorem of Royden-Earle-Kra, any isometry is induced by an element of the extended mapping class group. Moreover, any finite subgroup of the extended mapping class group has a fixed point in Teichmüller space. This is known as the Nielsen realization problem, solved by Kerckhoff.

Since the proofs for Teichmüller space are very specific to the geometry of surfaces, I suspect that the answer might be ''no'' in general. However, I don't know many examples of uniquely geodesic spaces besides the above. One way to construct new examples is to take $\ell^p$ products of uniquely geodesic spaces for $1<p<\infty$, but I don't think this would yield a counterexample.

Thus a subquestion is: What are good examples of uniquely geodesic spaces which are not Busemann spaces, Teichmüller spaces, or products thereof?

Suppose that a finite group $G$ acts on a proper (closed balls are compact) uniquely geodesic space $X$ by isometries. Then is there a point fixed by all of $G$?

This is true if $X$ is non-positively curved in the sense of Busemann (in particular for $CAT(0)$ spaces). Given any $y\in X$, the function $$ f(x) = \sum_{g\in G} d(x,gy)^2 $$ is proper and strictly convex, hence achieves its minimum at a unique point. The latter is fixed by $G$ since $f$ is invariant under $G$. (This is taken from Busemann's "The geometry of geodesics")

The result is also true when $X$ is Teichmüller space equipped with the Teichmüller metric. By a theorem of Royden-Earle-Kra, any isometry is induced by an element of the extended mapping class group. Moreover, any finite subgroup of the extended mapping class group has a fixed point in Teichmüller space. This is known as the Nielsen realization problem, solved by Kerckhoff.

Since the proofs for Teichmüller space are very specific to the geometry of surfaces, I suspect that the answer might be ''no'' in general. However, I don't know many examples of uniquely geodesic spaces besides the above. One way to construct new examples is to take $\ell^p$ products of uniquely geodesic spaces for $1<p<\infty$, but I don't think this would yield a counterexample.

Thus a subquestion is: What are good examples of uniquely geodesic spaces which are not Busemann spaces, Teichmüller spaces, or products thereof?

Edit: someone asked this question in the case G is cyclic here.