{'fraction_non_alphanumeric': 0.04234972677595628, 'fraction_numerical': 0.030054644808743168, 'mean_word_length': 4.470149253731344, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A connection between 5-systolicity and CAT ( − 1 ) metrics on cubical complexes

We investigate the existence of a CAT(−1) metric on a piecewise hyperbolic cubical complex of edge lengths ε whose cells represent only finitely many isometry classes, and satisfying a combinatorial curvature condition called local 5-largeness. We will show that, for some ε, the star of any cell of a locally 5-large cubical complex is locally a CAT(−1) space, whereas any 5-systolic cubical complex is a CAT(−1) space. The key step of our proof is to show that the link of any vertex of an 5-large regular piecewise hyperbolic cubical complex is a flag piecewise spherical simplicial cell complex with no empty 4-circuits. M.S.C. 2010: 37F20, 57Q15.