{'fraction_non_alphanumeric': 0.04257130693912303, 'fraction_numerical': 0.011068539804171988, 'mean_word_length': 4.353075170842825, '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}}
On the semisimplicity of polyhedral isometries

If a polyhedral complex K has only finitely many isometry types of cells, then all of its cellular isometries are semisimple. If K is 1-connected and non-positively curved, then any solvable group that acts freely by cellular isometries on K is finitely generated and contains an abelian subgroup of finite index. In this note I shall prove that if a metric polyhedral complex has only finitely many isometry types of cells, then all of its cellular isometries are semisimple (Theorem A). This theorem is a consequence of the results in the first chapter of my thesis [B], and I ought to have made it explicit at the time. While correcting this oversight, I shall give an application that indicates the utility of Theorem A. Given an isometry γ of a metric space X , let |γ| = inf{d(x, γ.x) | x ∈ X}. One says that γ is semisimple if there exists x0 ∈ X such that d(x0, γ.x0) = |γ|. Following [B], we write Shapes(K) to denote the set of isometry classes of cells in a metric polyhedral complex. Theorem A. Let K be a connected Mκ-polyhedral complex. If Shapes(K) is finite, then every cellular isometry of K is semisimple. Theorem B. If K is a 1-connected, non-positively curved, Mκ-polyhedral complex with Shapes(K) finite, and if Γ is a solvable group that acts freely by cellular isometries on K, then Γ is finitely generated and contains an abelian subgroup of finite index. (The torsion-free rank of this abelian subgroup is no greater than the dimension of K.) Theorem B does not extend to the case of semisimple actions on complete, 1connected, non-positively curved complexes in which Shapes(K) is not finite. For instance the additive rationals act freely and semisimply on a 2-dimensional complex of this type [BH, II.7]. In the case where K is a smooth manifold, Theorem B is due to Gromoll and Wolf [GW] (see also Lawson and Yau [LY]). Gromov [G2] was the first to point out that the results of [GW] remain valid beyond the setting of Riemannian manifolds. The elements of Theorems A and B can be found scattered in my book with André Haefliger [BH]. I was motivated to write this more concise and explicit account by an interesting lecture which Jacek Świa̧tkowski gave at the Banach Received by the editors October 7, 1997. 1991 Mathematics Subject Classification. Primary 53C23, 20F32.