arXiv subjects

Michael Mihalik

Publications and source records attributed to Michael Mihalik.

18 recordsLinked to original sources

arXiv2026

The higher connectivity at infinity of mapping class groups

The higher connectivity at infinity for mapping class groups of surfaces with boundary components and punctures is understood with the exceptions of the mapping class groups for the closed surfaces of genus 3 and 4. In this paper we prove a general simply connected at infinity result for finitely presented groups that implies all mapping class groups of closed surfaces of genus $\geq 3$ are simply connected at infinity. As these groups are duality groups the Proper Hurewicz Theorem implies that they are $(n-2)$-connected at infinity where $n$ is the dimension of the group. Combining this result with earlier work we give a complete list of all mapping class groups and their connectivity at infinity.

math.GR↗

arXiv2026

A Manual for Ends, Semistability and Simple Connectivity at Infinity for Groups and Spaces

This $2^{nd}$-edition 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 $H^2(G,\mathbb ZG)$ is free abelian or trivial. The idea is not to reprove these results, but to provide a historical record of the progress on these questions and provide a list of the most general results. We also prove or cite all of the results that make up the basic theory. The first Chapter is devoted to ends of groups and spaces, and the second to semistability at infinity, simple connectivity at infinity and second cohomology of groups. Definitions, basic facts and lists of general results are given in each Chapter. A number of results proven here are new and a number of authors have contributed results. We end with an Index for simply connected at infinity groups and an Index of Groups and Subgroups which is intended to help a reader quickly locate results about certain types of groups/subgroups. The main updates from the first edition is section 2.4.5 on mapping class groups and the addition of the simply connected at infinity index.

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arXiv2025

Splittings of One-Ended Groups with One-Ended Halfspaces

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 splitting are subgraphs of the Cayley graph obtained by pulling back the halfspaces of the Bass--Serre tree. Our first theorem shows that (under mild conditions) any splitting of a one-ended group can be upgraded to a splitting where all the halfspaces are one-ended. Our second theorem demonstrates that a one-ended group usually has a JSJ splitting where all the halfspaces are one-ended. And our third theorem states that if a one-ended finitely presented group $G$ admits a splitting such that some edge stabilizer has more than one end, but the halfspaces associated to the edge stabilizer are one-ended, then $H^2(G,\mathbb ZG)\ne \{0\}$; in particular $G$ is not simply connected at infinity and $G$ is not an $n$-dimensional duality group for $n\geq3$.

math.GR↗

arXiv2025

Local Connectivity of Right-angled Coxeter group boundaries

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 published paper where we streamline some of the proofs and add figures.

math.GR↗

arXiv2025

Stallings' Group is Simply Connected at Infinity

Let $F_2$ be the free group on two generators and let $B_n$ ($n \geq 2$) denote the kernel of the homomorphism $$F_2 \times \cdots (n) \cdots \times F_2 \rightarrow {\mathbb Z}$$ sending all generators to the generator $1$ of $\mathbb Z$. The groups $B_k$ are called the {\it Bieri-Stallings} groups and $B_k$ is type $\mathcal F_{k-1}$ but not $\mathcal F_k$. For $n\geq 3$ there are short exact sequences of the form $$1 \rightarrow B_{n-1} \rightarrow B_n \rightarrow F_2 \rightarrow 1.$$ This exact sequence can be used to show that $B_n$ is $(n-3)$-connected at infinity for $n\geq 3$. Stallings' proved that $B_2$ is finitely generated but not finitely presented. We conjecture that for $n\geq 2$, $B_n$ is $(n-2)$-connected at infinity. For $n=2$, this means that $B_2$ is 1-ended and for $n=3$ that $B_3$ (typically called Stallings' group) is simply connected at infinity. We verify the conjecture for $n=2$ and $n=3$. Our main result is the case $n=3$: Stalling's group is simply connected at $\infty$.

math.GR↗

arXiv2025

The Lamplighter Group is Not Semistable at Infinity

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 definition for finitely presented groups. At that time we suggest that the Lamplighter group may not be semistable at infinity, but until now there was no confirmed example of a finitely generated group that is not semistable at infinity. We prove the Lamplighter group is not semistable at infinity. Finitely generated non-semistable groups may be important in finding non-semistable finitely presented groups via ascending HNN extensions. There is an ascending HNN extension E of the Lamplighter group (called the Extended Lamplighter group) that is finitely presented. It would seem that E is a candidate to be a finitely presented non-semistable at infinity group, but a result of N. Silkin, shows that E is in fact simply connected at infinity.

math.GR↗

arXiv2022

Near Ascending HNN-Extensions and a Combination Result for Semistability at Infinity

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 presented groups have semistable fundamental group at infinity. While many classes of groups are known to contain only semistable at infinity groups, there are only a few combination results for such groups. Our main theorem is such a result. Main Theorem. Suppose $G$ is the fundamental group of a connected reduced graph of groups, where each edge group is infinite and finitely generated, and each vertex group is finitely presented and either 1-ended and semistable at infinity or has an edge group of finite index. Then $G$ is 1-ended and semistable at infinity. An important part of the proof of this result is the semistability part of the following: Theorem. Suppose $H_0$ is an infinite finitely presented group, $H_1$ is a subgroup of finite index in $H_0$, $ϕ:H_1\to H_0$ is a monomorphism and $G=H_0\ast_ϕ$ is the resulting HNN extension. Then $G$ is 1-ended and semistable at infinity. If additionally, $H_0$ is 1-ended, then $G$ is simply connected at infinity.

math.GR↗

arXiv2021

Relatively Hyperbolic Groups with Semistable Peripheral Subgroups

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 no cut point, then $G$ was known to have semistable fundamental group at $\infty$. We consider the more general situation when $\partial (G,{\bf P})$ contains cut points. Our main theorem states that if $G$ is finitely presented and each $P\in {\bf P}$ is finitely generated and has semistable fundamental group at $\infty$, then $G$ has semistable fundamental group at $\infty$.

math.GR↗

arXiv2020

Semistability of Graph Products

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 relations $[G_v,G_w]=1$ for all adjacent $v$ and $w$ in $Γ$. A finitely presented group $G$ has {\it semistable fundamental group at $\infty$} if for some (equivalently any) finite connected CW-complex $X$ with $π_1(X)=G$, the universal cover $\tilde X$ of $X$ has the property that any two proper rays in $\tilde X$ are properly homotopic. The class of finitely presented groups with semistable fundamental group at $\infty$ is known to contain many other classes of groups, but it is a 40 year old question as to whether or not all finitely presented groups have semistable fundamental group at $\infty$. Our main theorem is a combination result. It states that if $G$ is a graph product on a finite graph $Γ$ and each vertex group is finitely presented, then $G$ has non-semistable fundamental group at $\infty$ if and only if there is a vertex $v$ of $Γ$ such that $G_v$ is not semistable, and the subgroup of $G$ generated by the vertex groups of vertices adjacent to $v$ is finite (equivalently $lk(v)$ is a complete graph and each vertex group of $lk(v)$ is finite). Hence if one knows which vertex groups of $G$ are not semistable and which are finite, then an elementary inspection of $Γ$ determines whether or not $G$ has semistable fundamental group at $\infty$.

math.GR↗

arXiv2020

Relatively hyperbolic groups with free abelian second cohomology

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 $\partial (G,{\mathcal P})$ is locally connected and the second cohomology group $H^2(P,\mathbb ZP)$ is free abelian for each $P\in \mathcal P$, then $H^2(G,\mathbb ZG)$ is free abelian. When $G$ is 1-ended it is conjectured that $\partial (G,\mathcal P)$ is always locally connected. Under mild conditions on $G$ and the members of $\mathcal P$ the 1-ended and local connectivity hypotheses can be eliminated and the same conclusion is obtained. When $G$ and each member of $\mathcal P$ is 1-ended and $\partial (G,\mathcal P)$ is locally connected, we prove that the "Cusped Space" for this pair has semistable fundamental group at $\infty$. This provides a starting point in our proof of the main theorem.

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arXiv2018

Topological properties of spaces admitting a coaxial homeomorphism

Wright showed that, if a 1-ended simply connected locally compact ANR Y with pro-monomorphic fundamental group at infinity admits a proper Z-action, then that fundamental group at infinity can be represented by an inverse sequence of finitely generated free groups. Geoghegan and Guilbault strengthened that result, proving that Y also satisfies the crucial "semistability" condition. Here we get a stronger theorem with weaker hypotheses. We drop the pro-monomorphic hypothesis and simply assume that the Z-action is generated by what we call a "coaxial" homeomorphism. In the pro-monomorphic case every proper Z-action is generated by a coaxial homeomorphism, but coaxials occur in far greater generality (often embedded in a cocompact action). When the generator is coaxial, we obtain the sharp conclusion: Y is proper 2-equivalent to the product of a locally finite tree with a line. Even in the pro-monomorphic case this is new: it says that, from the viewpoint of fundamental group at infinity, the end of Y looks like the suspension of a totally disconnected compact set.

math.GT↗

arXiv2017

Non-cocompact Group Actions and $π_1$-Semistability at Infinity

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 homotopic. This property of $Y$ captures a notion of connectivity at infinity stronger than "1-ended", and is in fact a feature of $G$, being independent of choices. It is a fundamental property in the homotopical study of finitely presented groups. While many important classes of groups have been shown to have semistable fundamental group at infinity, the question of whether every $G$ has this property has been a recognized open question for nearly forty years. In this paper we attack the problem by considering a proper {\it but non-cocompact} action of a group $J$ on such an $Y$. This $J$ would typically be a subgroup of infinite index in the geometrically acting over-group $G$; for example $J$ might be infinite cyclic or some other subgroup whose semistability properties are known. We divide the semistability property of $G$ into a $J$-part and a "perpendicular to $J$" part, and we analyze how these two parts fit together. Among other things, this analysis leads to a proof (in a companion paper) that a class of groups previously considered to be likely counter examples do in fact have the semistability property.

math.GR↗

arXiv2017

Bounded Depth Ascending HNN Extensions and $π_1$-Semistability at $\infty$

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 properly homotopic. If $G$ has semistable fundamental group at $\infty$ then one can unambiguously define the fundamental group at $\infty$ for $G$. The problem, asking if all finitely presented groups have semistable fundamental group at $\infty$ has been studied for over 40 years. If $G$ is an ascending HNN extension of a finitely presented group then indeed, $G$ has semistable fundamental group at $\infty$, but since the early 1980's it has been suggested that the finitely presented groups that are ascending HNN extensions of {\it finitely generated} groups may include a group with non-semistable fundamental group at $\infty$. Ascending HNN extensions naturally break into two classes, those with bounded depth and those with unbounded depth. Our main theorem shows that bounded depth finitely presented ascending HNN extensions of finitely generated groups have semistable fundamental group at $\infty$. Semistability is equivalent to two weaker asymptotic conditions on the group holding simultaneously. We show one of these conditions holds for all ascending HNN extensions, regardless of depth. We give a technique for constructing ascending HNN extensions with unbounded depth. This work focuses attention on a class of groups that may contain a group with non-semistable fundamental group at $\infty$.

math.GR↗

arXiv2014

Semistability and Simple Connectivity at Infinity of Finitely Generated Groups with a Finite Series of Commensurated Subgroups

A subgroup $H$ of a group $G$ is $commensurated$ in $G$ if for each $g\in G$, $gHg^{-1}\cap H$ has finite index in both $H$ and $gHg^{-1}$. If there is a sequence of subgroups $H=Q_0\prec Q_1\prec ...\prec Q_{k}\prec Q_{k+1}=G$ where $Q_i$ is commensurated in $Q_{i+1}$ for all $i$, then $Q_0$ is $subcommensurated$ in $G$. In this paper we introduce the notion of the simple connectivity at infinity of a finitely generated group (in analogy with that for finitely presented groups). Our main result is: If a finitely generated group $G$ contains an infinite, finitely generated, subcommensurated subgroup $H$, of infinite index in $G$, then $G$ is 1-ended and semistable at $\infty$. If additionally, $H$ is finitely presented and 1-ended, then $G$ is simply connected at $\infty$. A normal subgroup of a group is commensurated, so this result is a strict generalization of a number of results, including the main theorems of G. Conner and M. Mihalik \cite{CM}, B. Jackson \cite{J}, V. M. Lew \cite{L}, M. Mihalik \cite{M1}and \cite{M2}, and J. Profio \cite{P}.

math.GR↗

arXiv2012

A classification of right-angled Coxeter groups with no 3-flats and locally connected boundary

If $(W,S)$ is a right-angled Coxeter system and $W$ has no $\mathbb Z^3$ subgroups, then it is shown that the absence of an elementary separation property in the presentation diagram for $(W,S)$ implies all CAT(0) spaces acted on geometrically by $W$ have locally connected CAT(0) boundary. It was previously known that if the presentation diagram of a general right-angled Coxeter system satisfied the separation property then all CAT(0) spaces acted on geometrically by $W$ have non-locally connected boundary. In particular, this gives a complete classification of the right-angled Coxeter groups with no 3-flats and with locally connected boundary.

math.GR↗

arXiv2009

JSJ Decompositions of Coxeter Groups

The idea of "JSJ-decompositions" for 3-manifolds began with work of Waldhausen and was developed later through work of Jaco, Shalen and Johansen. It was shown that there is a finite collection of 2-sided, incompressible tori that separate a given closed irreducible 3-manifold into pieces with strong topological structure. Sela introduced the idea of JSJ-decompositions for groups, an idea that has flourished in a variety of directions. The general idea is to consider a certain class X of groups and splittings of groups in X by groups in another class Y. E.g. Rips and Sela considered splittings of finitely presented groups by infinite cyclic groups. For an arbitrary group G in X the goal is to produce a unique graph of groups decomposition T of G with edge groups in Y so that T reveals all graph of groups decompositions of G with edge groups in Y. More specifically, if V is a vertex group of T then either there is no Y-group that splits both G and V, or V has a special "surface group-like" structure. It is standard to call vertex groups of the second type "orbifold groups". For a finitely generated Coxeter system (W,S) we produce a reduced JSJ-decomposition T for splittings of W over virtually abelian subgroups. We show T is unique with each vertex and edge group generated by a subset of S (and so T is "visual"). The construction of T is algorithmic. If V, a subset of S, generates an orbifold vertex group of T then V is the disjoint union of K and M, where < M > is virtually abelian, < K > is virtually a closed surface group or virtually free and < V > is the direct product of < M > and < K >.

math.GR↗

arXiv2006

Quotient isomorphism invariants of a finitely generated Coxeter group

In this paper we describe a family of isomorphism invariants of a finitely generated Coxeter group W. Each of these invariants is the isomorphism type of a quotient group W/N of W by a characteristic subgroup N. The virtue of these invariants is that W/N is also a Coxeter group. For some of these invariants, the isomorphism problem of W/N is solved and so we obtain isomorphism invariants that can be effectively used to distinguish isomorphism types of finitely generated Coxeter groups.

math.GR↗

arXiv2006

Matching theorems for systems of a finitely generated Coxeter group

In this paper we prove a series of matching theorems for two sets of Coxeter generators of a finitely generated Coxeter group that identify common features of the two sets of generators. As an application, we describe an algorithm for finding a set of Coxeter generators of maximum rank for a finitely generated Coxeter group.

math.GR↗