Let M be a compact manifold. We show the identity component Homeo0(M) of the group of self-homeomorphisms of M has a well-defined quasi-isometry type, and study its large scale geometry. Through examples, we relate this large scale geometry to both the topology of M and the dynamics of group actions on M . This gives a rich family of examples of non-locally compact groups to which one can apply...

Wederive ahomeomorphism invariant for those tiling spaceswhich aremadeby rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in in¢nitely manyorientations.The invariant is a quotient of C4 ech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheel-like tiling s...

1. R. B. Banks, Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics, Princeton University Press, Princeton, 1999. 2. A. C. Doyle, “The Stone of Boxman’s Drift,” in Uncollected Stories: The Unknown Conan Doyle, Doubleday, New York, 1982. 3. G. Shortley and D. Williams, Elements of Physics, Prentice-Hall, Englewood Cliffs, NJ, 1971. 4. A. J. Simoson, Falling down a hole ...

Let M be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of M has the automatic continuity property : any homomorphism from Homeo(M) to any separable group is necessarily continuous. This answers a question of C. Rosendal. If N ⊂ M is a submanifold, the group of homeomorphisms of M that preserve N also has this property. Various applications of automatic con...

A quasisymmetric homeomorphism defines an element of the universal Teichmüller space and a symmetric one belongs to its little subspace. We show example h real line \({\mathbb {R}}\) onto itself such that \(h^{-1}\) is not symmetric. This implies set all self-homeomorphisms does constitute group under composition. also consider same problem for strongly self-homeomorphism which defined by certa...

Radial lines, suitably parameterized, are geodesics, but notice that the distance from the origin to the (Euclidean) unit sphere is infinite. This model makes it intuitively clear that the boundary at infinity of hyperbolic space is Sn−1. Hyperbolic space together with its boundary at infinity has the topology of a closed ball, and isometries of hyperbolic space extend uniquely to a homeomorphi...

Definition For an element x of a topological space X , we will write Ux for the set of all open neigborhoods of x. f : X → Y is a local homeomorphism if for every x ∈ X there exists a U ∈ Ux such that f |U is a homeomorphism onto an open subset of Y . Denote by intA the interior of A ⊆ X . For any pair of maps φ : U → Y , φ : V → Y defined on open subsets U, V ⊆ X , define the (set) equalizer e...

The Thompson group V is a subgroup of the homeomorphism group of the Cantor set C. Brin [3] defined higher dimensional Thompson groups nV as generalizations of V . For each n, nV is a subgroup of the homeomorphism group of Cn. We prove that the number of ends of nV is equal to 1, and there is a subgroup of nV such that the relative number of ends is ∞. This is a generalization of the correspond...