Small-cancellation groups with and without sigma-compact Morse boundary

Abstract

We provide examples of classical $C^{\prime }(1/6)$-small-cancellation groups which have non-$\sigma$-compact Morse boundary. These are first known examples of groups with non-$\sigma$-compact Morse boundary. Some $C^{\prime }(1/6)$-small-cancellation groups do have $\sigma$-compact Morse boundary, so this property distinguishes quasi-isometry types of small-cancellation groups. In fact, we give a complete description of when Morse boundaries of $C^{\prime }(1/6)$-groups have $\sigma$-compact Morse boundary. We also provide examples of $C^{\prime }(1/6)$-groups where all Morse rays are strongly contracting.