arXiv research

New proof shows abelian Cantor groups can act on spaces.

The group of $\mathcal C^1$-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson's groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin's higher dimensional generalizations $nV$ of Thompson's group $V$ arise…

By a Cantor group we mean a topological group homeomorphic to the Cantor set. We show that a compact metric space of rational cohomological dimension $n$ can be obtained as the orbit space of a Cantor group action on a metric compact space of covering dimension $n$. Moreover, the action can be assumed to be free if $n=…

In this article, we study the smooth mapping class group of a surface S relative to a given Cantor set, that is the group of isotopy classes of orientation-preserving smooth diffeomorphisms of S which preserve this Cantor set. When the Cantor set is the standard ternary Cantor set, we prove that the subgroup consisting…

For each Cantor set C in $R^{3}$, all points of which have bounded local genus, we show that there are infinitely many inequivalent Cantor sets in $R^{3}$ with complement having the same fundamental group as the complement of C. This answers a question from Open Problems in Topology and has as an application a simple c…

For every finitely generated abelian group G, we construct an irreducible open 3-manifold $M_{G}$ whose end set is homeomorphic to a Cantor set and with end homogeneity group of $M_{G}$ isomorphic to G. The end homogeneity group is the group of self-homeomorphisms of the end set that extend to homeomorphisms of the 3-m…

Every normal subgroup of Cantor tree's mapping class group is geometric.

Study shows similar result to Margulis for Cantor set homeomorphisms.

The study classifies normal subgroups of mapping class groups of surfaces with Cantor subsets.

We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension $<1$ are free. On the other hand we construct for any $ε>0$ examples of non-free purely hyperbolic Kleinian groups whose limit set is a Cantor set of Hausdorff dimension $<1+ε$.

We study direct limits of embedded Cantor sets and embedded \sier curves. We show that under appropriate conditions on the embeddings, all limits of Cantor spaces give rise to homeomorphic spaces, called $ω$-Cantor spaces, and similarly, all limits of \sier curves give homeomorphic spaces, called to $ω$-\sier curves. W…

Proves asymptotic mapping class groups of Cantor manifolds are of type F_infinity.

Proves homology of mapping class groups for infinite-type surfaces.

The study explores homeomorphism groups of self-similar 2-manifolds, including the 2-sphere and Cantor set.

In this paper, for a non compact and orientable surface $S$ been either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we construct explicitly an infinitely generated Fuchsian group $Γ<PSL(2,\mathbb{R})$, such that the quotient $\mathbb{H}/Γ$ is a hyperbolic surface homeomorphic to $S$.

Uncountably many fibrations found on genus 2 handlebody.

Groups acting on product trees are boundary rigid.

The paper proves identities for hyperconvex Anosov representations and their applications to Cantor sets.

Infinite clique of rays in plane minus Cantor set.

Study shows certain mapping class groups cannot be realized as subgroup of homeomorphisms.

Perfect mapping class groups of specific surfaces have no proper subgroups.

Study of SL(2,R) representations on a once-punctured torus, showing Cantor set spectrum.

We will show that if a proper complete CAT(0) space X has a visual boundary homeomorphic to the join of two Cantor sets, and X admits a geometric group action by a group containing a subgroup isomorphic to Z^2, then its Tits boundary is the spherical join of two uncountable discrete sets. If X is geodesically complete,…

The paper explores low-dimensional solenoidal manifolds and their properties.

We specify exactly which groups can act geometrically on CAT(0) spaces whose visual boundary is homeomorphic to either a circle or a suspension of a Cantor set.

Constructs infinitely many non-equivalent wild knots in Menger sponge.

We introduce a topological object, called hairy Cantor set, which in many ways enjoys the universal features of objects like Jordan curve, Cantor set, Cantor bouquet, hairy Jordan curve, etc. We give an axiomatic characterisation of hairy Cantor sets, and prove that any two such objects in the plane are ambiently homeo…

The ray graph is a Gromov hyperbolic graph on which the mapping class group of the plane minus a Cantor set acts by isometries. We give a description of the Gromov boundary of the ray graph in terms of cliques of long rays on the plane minus a Cantor set. As a consequence, we prove that the Gromov boundary of the ray g…

Free groups can be end homogeneity groups of 3-manifolds.

We found a new simple family of Cantor sets whose projections are one-dimensional.

In the paper, we provide an effective method for the Lipschitz equivalence of two-branch Cantor sets and three-branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent if and only if their contraction vectors are equivalent provided one of the cont…

Bing-Whitehead Cantor sets were introduced by DeGryse and Osborne in dimension three and greater to produce examples of Cantor sets that were non standard (wild), but still had simply connected complement. In contrast to an earlier example of Kirkor, the construction techniques could be generalized to dimensions bigger…

First constructed genus 2 Cantor set in 3D space.

Study homeomorphism groups of ordinals, proving strong distortion and normal generators.

We present the characterization of metric spaces that are micro-, macro- or bi-uniformly equivalent to the extended Cantor set $\{\sum_{i=-n}^\infty\frac{2x_i}{3^i}:n\in\IN ,\;(x_i)_{i\in\IZ}\in\{0,1\}^\IZ\}\subset\IR$, which is bi-uniformly equivalent to the Cantor bi-cube $2^{<\IZ}=\{(x_i)_{i\in\IZ}\in \{0,1\}^\IZ:\e…

All projections of typical Cantor sets in high dimensions are Cantor sets.

Let $Γ$ denote the mapping class group of the plane minus a Cantor set. We show that every action of $Γ$ on the circle is either trivial or semi-conjugate to a unique minimal action on the so-called simple circle.

Cantor Riemannium is a new type of space from holomorphic germs.

Non-ergodic geodesic flow on Cantor tree surfaces found.

New Cantor sets with high-dimensional projections discovered.

Hyperbolic 3-manifolds can be approximated by removing Cantor sets from the 3-sphere.

Researchers create a Fredholm module on fractal shapes like the Cantor set.

A classical theorem of Alexandroff states that every $n$-dimensional compactum $X$ contains an $n$-dimensional Cantor manifold. This theorem has a number of generalizations obtained by various authors. We consider extension-dimensional and infinite dimensional analogs of strong Cantor manifolds, Mazurkiewicz manifolds,…

We construct a Cantor set in S^3 whose complement admits a complete hyperbolic metric.

The study confirms most Cantor sets are in general position for all projections.

We introduce subgroups ${\mathcal{B}}_g< {\mathcal H}_g$ of the mapping class group $Mod(Σ_g)$ of a closed surface of genus $g \ge 0$ with a Cantor set removed, which are extensions of Thompson's group $V$ by a direct limit of mapping class groups of compact surfaces of genus $g$. We first show that both ${\mathcal{B}}…

New minimal surfaces found with Cantor ends in convex domains.

We construct a large class of pathological $n$-dimensional topological spheres in ${\mathbb R}^{n+1}$ by showing that for any Cantor set $C\subset {\mathbb R}^{n+1}$ there is a topological embedding $f:{\mathbb S}^n\to{\mathbb R}^{n+1}$ of the Sobolev class $W^{1,n}$ whose image contains the Cantor set $C$.