In this paper we generalize a result in [J. An, Z. Wang, On the realization of Riemannian symmetric spaces in Lie groups, Topology Appl. 153 (7) (2005) 10081015, showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of fixed points of involutions are also proved.

The aim of this paper is to give a proof of the calculation of the flabby class group of a finite cyclic group due to Endo and Miyata [2]. In the next section I will recall the definition and some basic facts about this group. In the final section I will give some examples to show that the invertibility conditions used by Endo and Miyata cannot be removed. I would like to thank M.c. Kang for useful comments and for showing me his results on some related problems.

Let (W, I) be a finite Coxeter group. In the case where W is a Weyl group, Berenstein and Kazhdan in [A. Berenstein, D. Kazhdan, Geometric and unipotent crystals. II. From unipotent bicrystals to crystal bases, in: Quantum Groups, in: Contemp. Math., vol. 433, Amer. Math. Soc., Providence, RI, 2007, pp. 1388] constructed a monoid structure on the set of all subsets of I using unipotent -linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type.

(1) Ker (f,) is a characteristic subgroup of T,(M),(2) the group G leaves Ker (f*) invariant. This assumption is needed in the last two statements of Corollary 6,(vi) of Theorems 8, Theorem 11 and Theorem 13. In Theorem 12, the corresponding assumption can be stated as follows: Let H be the subspace of H,(M, I?) defined by {PIP U ai, U... U ai, _,= 0 for all ii}. Then G leaves H invariant.

We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with exotic properties: • We construct the first examples of finitely generated coarsely non-amenable groups (that is, groups without Guoliang Yu’s Property A) that are coarsely embeddable into a Hilbert space. Moreover, our groups act properly on CAT(0) cubical complexes. • We construct the first examples of finitely generated groups, with expanders embedded isometrically into their Cayley graphs—in contrast, in the case of the Gromov monster expanders are not even coarsely embedded. We present further applications.