On Homeomorphism Groups of Menger Continua

On the Topology of Grid Continua

One-dimensional and two-dimensional continua belong to the basic notions of settheoretical topology and represent a subfield of the theory of dimensions developed by P. Urysohn and K. Menger. In this paper basic definitions and properties of grid continua in R2 and R3 are summarised. Particularly, simple one-dimensional grid continua in R2 and in R3, and simple closed two-dimensional grid conti...

Topology Proceedings 7 (1982) pp. 391-393: QUESTIONS ON HOMEOMORPHISM GROUPS OF CHAINABLE AND HOMOGENEOUS CONTINUA

On indecomposability and composants of chaotic continua

A homeomorphism f : X → X of a compactum X with metric d is expansive if there is c > 0 such that if x, y ∈ X and x 6= y, then there is an integer n ∈ Z such that d(fn(x), fn(y)) > c. A homeomorphism f : X → X is continuum-wise expansive if there is c > 0 such that if A is a nondegenerate subcontinuum of X, then there is an integer n ∈ Z such that diam fn(A) > c. Clearly, every expansive homeom...

Hyperbolic groups with 1-dimensional boundary

If a torsion-free hyperbolic group G has 1-dimensional boundary ∂∞G, then ∂∞G is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When ∂∞G is a Sierpinski carpet we show that G is a quasiconvex subgroup of a 3-dimensional hyperbolic Poincaré duality group. We also construct a “topologically rigid” hyperbolic group G: any homeomorphism of ∂∞G is induced by an ...

The nonexistence of expansive homeomorphisms of chainable continua

A homeomorphism f : X → X of a compactum X with metric d is expansive if there is c > 0 such that if x, y ∈ X and x 6= y, then there is an integer n ∈ Z such that d(f(x), f(y)) > c. In this paper, we prove that if a homeomorphism f : X → X of a continuum X can be lifted to an onto map h : P → P of the pseudoarc P , then f is not expansive. As a corollary, we prove that there are no expansive ho...