Michael Mihalik
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View article: Local Connectivity of Right-angled Coxeter group boundaries
Local Connectivity of Right-angled Coxeter group boundaries Open
We provide conditions on the defining graph of a right-angled Coxeter group presentation that guarantees the boundary of any CAT(0) space on which the group acts geometrically will be locally connected. This is a revised version of a publi…
View article: A Manual for Ends, Semistability and Simple Connectivity at Infinity for Groups and Spaces
A Manual for Ends, Semistability and Simple Connectivity at Infinity for Groups and Spaces Open
This article is intended to be an up to date archive of the current state of the questions: Which finitely generated groups $G$: have semistable fundamental group at infinity; are simply connected at infinity; are such that $H^2(G,\mathbb …
View article: The Lamplighter Group is Not Semistable at Infinity
The Lamplighter Group is Not Semistable at Infinity Open
The question of whether or not all finitely presented groups are semistable at infinity has been studied for over 40 years. In 1986, we defined what it means for a finitely generated group to be semistable at infinity - in analogy with the…
View article: Splittings of One-Ended Groups with One-Ended Halfspaces
Splittings of One-Ended Groups with One-Ended Halfspaces Open
We introduce the notion of halfspaces associated to a group splitting, and investigate the relationship between the coarse geometry of the halfspaces and the coarse geometry of the group. Roughly speaking, the halfspaces of a group splitti…
View article: Lifting Semistability in Finitely Generated Ascending HNN-Extensions
Lifting Semistability in Finitely Generated Ascending HNN-Extensions Open
If a finitely generated group maps epimorphically onto a group , we are interested in the question: When does the semistability of imply is semistable? In this paper, we give an answer within the class of ascending HNN-extensions. More …
View article: Near Ascending HNN-Extensions and a Combination Result for Semistability at Infinity
Near Ascending HNN-Extensions and a Combination Result for Semistability at Infinity Open
Semistability at infinity is an asymptotic property of finitely presented groups that is needed in order to effectively define the fundamental group at infinity for a 1-ended group. It is an open problem whether or not all finitely present…
View article: Relatively Hyperbolic Groups with Semistable Peripheral Subgroups
Relatively Hyperbolic Groups with Semistable Peripheral Subgroups Open
Suppose $G$ is a finitely presented group that is hyperbolic relative to ${\bf P}$ a finite collection of 1-ended finitely generated proper subgroups of $G$. If $G$ and the ${\bf P}$ are 1-ended and the boundary $\partial (G,{\bf P})$ has …
View article: Piecewise visual, linearly connected metrics on boundaries of relatively hyperbolic groups
Piecewise visual, linearly connected metrics on boundaries of relatively hyperbolic groups Open
Suppose a finitely generated group [Formula: see text] is hyperbolic relative to [Formula: see text] a set of proper finitely generated subgroups of [Formula: see text]. Established results in the literature imply that a “visual” metric on…
View article: Semistability of Graph Products
Semistability of Graph Products Open
A {\it graph product} $G$ on a graph $Γ$ is a group defined as follows: For each vertex $v$ of $Γ$ there is a corresponding non-trivial group $G_v$. The group $G$ is the quotient of the free product of the $G_v$ by the commutation relation…
View article: Relatively hyperbolic groups with free abelian second cohomology
Relatively hyperbolic groups with free abelian second cohomology Open
Suppose $G$ is a 1-ended finitely presented group that is hyperbolic relative to $\mathcal P$ a finite collection of 1-ended finitely presented proper subgroups of $G$. Our main theorem states that if the boundary $\partial (G,{\mathcal P}…
View article: Bounded Depth Ascending HNN Extensions and $\pi_1$-Semistability at $\infty$
Bounded Depth Ascending HNN Extensions and $\pi_1$-Semistability at $\infty$ Open
A 1-ended finitely presented group has semistable fundamental group at $\infty$ if it acts geometrically on some (equivalently any) simply connected and locally finite complex $X$ with the property that any two proper rays in $X$ are prope…
View article: Bounded Depth Ascending HNN Extensions and $π_1$-Semistability at $\infty$
Bounded Depth Ascending HNN Extensions and $π_1$-Semistability at $\infty$ Open
A 1-ended finitely presented group has semistable fundamental group at $\infty$ if it acts geometrically on some (equivalently any) simply connected and locally finite complex $X$ with the property that any two proper rays in $X$ are prope…
View article: Non-cocompact Group Actions and $\pi_1$-Semistability at Infinity
Non-cocompact Group Actions and $\pi_1$-Semistability at Infinity Open
A finitely presented 1-ended group $G$ has {\\it semistable fundamental group\nat infinity} if $G$ acts geometrically on a simply connected and locally\ncompact ANR $Y$ having the property that any two proper rays in $Y$ are\nproperly homo…
View article: Non-cocompact Group Actions and $π_1$-Semistability at Infinity
Non-cocompact Group Actions and $π_1$-Semistability at Infinity Open
A finitely presented 1-ended group $G$ has {\it semistable fundamental group at infinity} if $G$ acts geometrically on a simply connected and locally compact ANR $Y$ having the property that any two proper rays in $Y$ are properly homotopi…
View article: Semistability and simple connectivity at ∞ of finitely generated groups with a finite series of commensurated subgroups
Semistability and simple connectivity at ∞ of finitely generated groups with a finite series of commensurated subgroups Open
A subgroup [math] of a group [math] is commensurated in [math] if for each [math] , [math] has finite index in both [math] and [math] . If there is a sequence of subgroups [math] where [math] is commensurated in [math] for all [math] , the…