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We give a criterion for group elements to have fixed points with respect to a semi-simple action on a complete CAT(0) space of finite topological dimension. As an application, we show that Thompson's group T and various generalizations of Thompson's group V have global fixed points when they act semi-simply on finite-dimensional complete CAT(0) spaces, while it is known that T and V act properly on infinite-dimensional CAT(0) cube complexes.

In this paper, we construct embeddings of right-angled Artin groups into higher dimensional Thompson groups. In particular, we embed every right-angled Artin groups into n-dimensional Thompson groups, where n is the number of complementary edges in the defining graph. It follows that Z n * Z embeds into nV for every n ≥ 1. IntoductionThe Thompson group V is an infinite simple finitely presented group, which is described as a subgroup of the homeomorphism group of the Cantor set C. Brin [1] defined higher dimensional Thompson groups as generalizations of the Thompson group V = 1V . By definition, n-dimensional Thompson group n 1 V embeds into n 2 V when n 1 ≤ n 2 . Brin [1] showed that V and 2V are not isomorphic. Bleak and Lanoue [3] showed n 1 V and n 2 V are isomorphic if and only if n 1 = n 2 .In [4], Bleak and Salazar-Díaz proved that Z 2 * Z does not embed in V . Recently, Corwin and Haymaker [6] determined which right-angled Artin groups embed into V . Using the nonembedding result of [4], they showed that Z 2 * Z is the only obstruction for a right-angled Artin group to be embedded into V . On the other hand, Belk, Bleak and Matucci [2] proved that a rightangled Artin group embeds in nV with sufficiently large n. They took n to be the sum of the number of vertices and the number of complementary edges in the defining graph. They conjectured that a right-angled Artin group embeds into (n − 1)V if and only if the right-angled Artin group does not contain Z n * Z . Corwin [5] constructed embeddings of Z n * Z into nV for every n ≥ 2. It follows that every nV with n ≥ 2 does not embed into V .In this paper, we give another construction of embeddings of right-angled Artin groups into higher-dimensional Thompson groups. In particular, we may embed a right-angled Artin group into nV , where n is the number of 1

It is conjectured that the central quotient of any irreducible Artin–Tits group is either virtually cyclic or acylindrically hyperbolic. We prove this conjecture for Artin–Tits groups that are known to be CAT(0) groups by a result of Brady and McCammond, that is, Artin–Tits groups associated with graphs having no 3-cycles and Artin–Tits groups of almost large type associated with graphs admitting appropriate directions. In particular, the latter family contains Artin–Tits groups of large type associated with cones over square-free bipartite graphs.

In this paper, we study isometric actions on finite-dimensional CAT(0) spaces for the Higman–Thompson groups $$T_n$$ T n , which are generalizations of Thompson’s group T. It is known that every semi-simple action of T on a complete CAT(0) space of finite covering dimension has a global fixed point. After this result, we show that every semi-simple action of $$T_n$$ T n on a complete CAT(0) space of finite covering dimension has a global fixed point. In the proof, we regard $$T_n$$ T n as ring groups of homeomorphisms of $$S^1$$ S 1 introduced by Kim, Koberda and Lodha, and use general facts on these groups.