For a random walk on a finitely generated group G we obtain a generalization of a classical inequality of Ancona. We deduce as a corollary that the identity map on G extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This provides new results for Martin compactifications of relatively hyperbolic groups.

We study large groups of birational transformations Bir(X), where X is a variety of dimension at least 3, defined over C or a subfield of C. Two prominent cases are when X is the projective space P^n, in which case Bir(X) is the Cremona group of rank n, or when X⊂P^{n+1} is a smooth cubic hypersurface. In both cases, and more generally when X is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from Bir(X) to Z/2, showing in particular that the group Bir(X) is not perfect, and thus not simple. As a consequence, we also obtain that the Cremona group of rank n⩾3 is not generated by linear and Jonquières elements.

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.

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.

(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.