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Register a new userSurface-like boundaries of hyperbolic groups
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We classify the boundaries of hyperbolic groups that have enough quasiconvex codimension-1 surface subgroups with trivial or cyclic intersections.
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We give a short proof of Masbaum and Reids result that mapping class groups involve any finite group, appealing to free quotients of surface groups and a result of Gilman, following Dunfield-Thurston.
Coning off a collection of uniformly quasiconvex subsets of a Gromov hyperbolic space leaves a new space, called the cone-off. Kapovich and Rafi generalized work of Bowditch to show this space is still Gromov hyperbolic. We show that the Gromov boundary of cone-off embeds in the boundary of the original hyperbolic space. (A stronger version of this result was previously obtained by Dowdall and Taylor; see Note in text.) Moreover, under some acylindricity assumptions we give a precise description of the image. As an application, we are able to characterize the elliptic and loxodromic elements of groups acting on certain cone-offs of acylindrical actions.
We prove continuity results for abstract epimorphisms of locally compact groups onto finitely generated groups.
Given a $T_0$ paratopological group $G$ and a class $mathcal C$ of continuous homomorphisms of paratopological groups, we define the $mathcal C$-$semicompletion$ $mathcal C[G)$ and $mathcal C$-$completion$ $mathcal C[G]$ of the group $G$ that contain $G$ as a dense subgroup, satisfy the $T_0$-separation axiom and have certain universality properties. For special classes $mathcal C$, we present some necessary and sufficient conditions on $G$ in order that the (semi)completions $mathcal C[G)$ and $mathcal C[G]$ be Hausdorff. Also, we give an example of a Hausdorff paratopological abelian group $G$ whose $mathcal C$-semicompletion $mathcal C[G)$ fails to be a $T_1$-space, where $mathcal C$ is the class of continuous homomorphisms of sequentially compact topological groups to paratopological groups. In particular, the group $G$ contains an $omega$-bounded sequentially compact subgroup $H$ such that $H$ is a topological group but its closure in $G$ fails to be a subgroup.
Given a group $X$ we study the algebraic structure of the compact right-topological semigroup $lambda(X)$ consisting of maximal linked systems on $X$. This semigroup contains the semigroup $beta(X)$ of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup $lambda(X)$ in the semigroup of all self-maps of the power-set of $X$ and using this representation describe the structure of minimal ideal and minimal left ideals of $lambda(X)$ for each twinic group $X$. The class of twinic groups includes all amenable groups and all groups with periodic commutators but does not include the free group with two generators.
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