Hyperbolic spaces not quasi-isometric to curve complexes

Key Points

• To determine a specific condition that prevents a hyperbolic space from being quasi-isometric to the curve complex of non-sporadic surfaces.
• Identification of characteristics related to hyperbolic spaces and their metrics.
• Application of findings to various hyperbolic complexes including arc complexes, disk complexes, and pants complexes.
• Analysis of free splitting complexes of free groups.
• Establishment of a definitive condition for non-quasi-isometry with curve complexes for non-sporadic surfaces.
• This condition applies broadly across different hyperbolic complexes, underlining the limits of quasi-isometric relations.