An element f of a group G is reversible if it is conjugated in G to its own inverse. It is plain that two elements f and h satisfying the relation hf h-1 = f-1 generate a, possibly nonfaithful, BS(1,-1)-action, where BS(1,-1) is the Baumslag-Solitar group (a, b bab-1 = a-1) also referred to as the Klein bottle group. Our main result is that any reversible interval exchange transformation (IET) is reversible by a finite order IET. Its proof uses a dynamical decomposition for BS(1,-1)-actions by IET which involves some specific IET-subgroups for which we give a precise description of the reversible elements. We also indicate consequences on the IET-realizations of a group containing reversible maps.
Let F be a hyperbolic group and G be the isometry group of a Gromov-hyperbolic and geodesic metric space. We study the action of the outer automorphism group Out(F) on the set X(F, G) of conjugacy classes of representations of F into G. For some Aut(F)-invariant subset A C F, we study the properties of A-stable representations which are generalizations of primitive stable representations defined by Minsky. Their conjugacy classes form Out(F)-invariant subsets of X(F, G) which contains the set of conjugacy classes of quasi-isometrically embedded representations. We establish a sufficient condition for the induced action to be properly discontinuous and use it when A is the set of commutators of F. Working with the derived subgroup [F, F], we also prove that [F, F]-stability is equivalent to the [F, F]- well-displacing property. Focusing on representations in PSL2(C), we find new characterizations of convex cocompact subgroups of PSL2(C).
For each pair (m, n) of positive integers with (m, n) (1, 1) and an arbitrary field F with algebraic closure F, let Polyd degrees m n (F) denote the space of m-tuples (f1(z), ... , fm(z)) E F [z]m of monic polynomials with coefficients in F of the same degree d such that the polynomials {fk(z)}mkD1 have no common root in F of multiplicity >= n. These spaces Polyd degrees m n (F) were first defined and studied by B. Farb and J. Wolfson as generalizations of spaces first studied by Arnold, Vassiliev and Segal and others in several different contexts. In a previous paper we determined explicitly the homotopy type of this space when F = C. Here we investigate this space when F = R.
In the first part of this work, we study Dirichlet-Voronoi domains for discrete isometry groups of Riemannian manifolds, in view of constructing cell structures on homogeneous (complete, real) flag manifolds, equivariant with respect to the action of the Weyl group. We give general results, allowing us to build such a structure from an admissible one on the domain. In particular, the injectivity radius plays a key role in the method. The second part starts with the computation of the injectivity radius of (real and complex) flag manifolds; a first step towards the application of the method developed in the first part. Then, with the help of the quaternion algebra, we investigate the particular case of the flag manifold O(3)/O(1)3 of SL3(R): we prove that the results of the first part apply and derive a new C53-equivariant cell structure on it, whose cellular complex of Z[C53]-modules is determined.
Let X and Y be closed G-manifolds and B C Y a closed invariant nonempty subset where G is a finite group. For any G-map f: X-* Y and for every subgroup H <= G, we introduce a Nielsen type number N (f H, BH) which is a lower bound for the number of connected components of WH-orbits of (f H )-1(BH). This theory generalizes existing Nielsen type numbers for various G and B with an application to the Nielsen Borsuk-Ulam theory for the minimal number of coincidences of f (x) = ft (x) where f : X-* Y and t a free involution on X.
