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A geodesic in a proper CAT(0) space is said to be rank 1 if it does not bound a flat half-plane and zero-width if it does not bound a flat strip of any width. Let $X$ be a geodesically complete CAT(0) space contains a rank 1 geodesic. Assume it admits a properly discontinuous cocompact action by a group $G$.

Under these conditions we know:

1)$X$ contains a rank 1 geodesic which is an axis of an isometry in $G$ ([Link, Lemma 4.2] https://arxiv.org/pdf/1706.00402.pdf).

2. $X$ contains zero-width geodesic ([Ricks, Theorem 2] https://arxiv.org/abs/1410.3921)

My question: Must $X$ necessarily contain a zero width geodesic which is the axis of some isometry in $G$?

A geodesic in a proper CAT(0) space is said to be rank 1 if it does not bound a flat half-plane and zero-width if it does not bound a flat strip of any width. Let $X$ be a geodesically complete CAT(0) space that contains a rank 1 geodesic. Assume it admits a properly discontinuous cocompact action by a group $G$.

Under these conditions we know:

1. $X$ contains a rank 1 geodesic which is an axis of an isometry in $G$ ([Link, Lemma 4.2] https://arxiv.org/pdf/1706.00402.pdf).

2. $X$ contains zero-width geodesic ([Ricks, Theorem 2] https://arxiv.org/abs/1410.3921).

My question: Must $X$ necessarily contain a zero-width geodesic which is the axis of some isometry in $G$?

A geodesic in a proper CAT(0) space is said to be rank 1 if it does not bound a flat half-plane and zero-width if it does not bound a flat strip of any width. Let $X$ be a geodesically complete CAT(0) space which is rank 1, i.e. one that contains a geodesic which does not bound a flat half-plane. Assume it admits a properly discontinuous cocompact action by a group $G$.

Under these conditions we know:

1)$X$ contains a rank 1 geodesic which is an axis of an isometry in $G$ ([Link, Lemma 4.2] https://arxiv.org/pdf/1706.00402.pdf).

2. $X$ contains zero-width geodesic ([Ricks, Theorem 2] https://arxiv.org/abs/1410.3921)

My question: Must $X$ necessarily contain a zero width geodesic which is the axis of some isometry in $G$?

A geodesic in a proper CAT(0) space is said to be rank 1 if it does not bound a flat half-plane and zero-width if it does not bound a flat strip of any width. Let $X$ be a geodesically complete CAT(0) space contains a rank 1 geodesic. Assume it admits a properly discontinuous cocompact action by a group $G$.

Under these conditions we know:

1)$X$ contains a rank 1 geodesic which is an axis of an isometry in $G$ ([Link, Lemma 4.2] https://arxiv.org/pdf/1706.00402.pdf).

2. $X$ contains zero-width geodesic ([Ricks, Theorem 2] https://arxiv.org/abs/1410.3921)

My question: Must $X$ necessarily contain a zero width geodesic which is the axis of some isometry in $G$?