Quasi-redirecting boundaries of groups with linear divergence and 3-manifold groups

Hoang Thanh Nguyen Hoang Thanh Nguyen, Department of Mathematics, FPT University, Hoa Hai ward, Ngu Hanh Son district, Da Nang, Vietnam nthoang.math@gmail.com

Abstract.

The quasi-redirecting (QR) boundary, introduced by Qing and Rafi, generalizes the Gromov boundary for studying the large-scale geometry of finitely generated groups. Although it is not known to exist for all such groups, its existence has been established for several important classes. We prove that if a finitely generated group $G$ has linear divergence, then its QR-boundary is well-defined and consists of a single point. In addition, we show that all finitely generated 3-manifold groups admit well-defined QR-boundaries.

Key words and phrases:

linear divergence, quasi-redirecting boundary, 3-manifold groups

2010 Mathematics Subject Classification:

20F65, 20F67

1. Introduction

The quasi-redirecting (QR) boundary is a close generalization of the Gromov boundary to all finitely generated groups [QR24]. One of the advantages of the QR-boundary is that it is a new quasi-isometry invariant boundary that is often compact, containing sublinearly Morse boundaries [QRT22], [QRT24] as topological subspaces, capturing a richer spectrum of hyperbolic-like behaviors, making it a promising new tool in geometric group theory.

The QR boundary is defined as follows:

Definition 1.1.

Let $\alpha,\beta\colon[0,\infty)\to X$ be two quasi-geodesic rays in a metric space $X$ . We say $\alpha$ can be quasi-redirected to $\beta$ (and write $\alpha\preceq\beta$ ) if there exists a pair of constants $(q,Q)$ such that for every $r>0$ , there exists a $(q,Q)$ –quasi-geodesic ray $\gamma$ that is identical to $\alpha$ inside the ball $B(\alpha(0),r)$ and eventually $\gamma$ becomes identical to $\beta$ . We say $\alpha\sim\beta$ if $\alpha\preceq\beta$ and $\beta\preceq\alpha$ .

The resulting set of equivalence classes forms a poset, denoted by $P(X)$ (in this paper, we will call it the QR-poset). This poset $P(X)$ , when equipped with a “cone-like topology” (see [QR24, Section 5]), is called the quasi-redirecting boundary (QR boundary) of $X$ and denoted by $\partial_{*}X$ .

Qing and Rafi [QR24] established key properties of the QR-boundary, with further developments in [GQV24]. While QR boundaries are shown to be well-defined for several classes of groups of interest, including relatively hyperbolic groups, Croke-Kleiner admissible groups, non-geometric 3-manifold groups [QR24], [NQ25], its existence for all finitely generated groups remains an open question.

Question 1.2.

[QR24, Question D] Let $X$ be a Cayley graph of a finitely generated group. Is $\partial_{*}X$ always defined? Is $\partial_{*}X$ always compact?

In [QR24, Section 4], the authors show that the QR-boundary of a direct product of two infinite finitely generated groups consists exactly of one point. They also mention the work in [McM] where the author shows the same holds for Baumslag-Solitar group. These groups share a common feature: linear divergence, a quasi-isometry invariant that measures the minimal path length outside a ball connecting two points on its boundary, as a function of the ball’s radius [Gro96], [Ger94a], [DMS10] (see the precise definition in Section 2.3).

This observation naturally raises the following question:

Question 1.3.

For a finitely generated group $G$ , does linear divergence imply that $\partial_{*}G$ is a single point?

Our result provides an affirmative answer to Question 1.3.

Theorem 1.4.

Let $G$ be a finitely generated group. If $G$ has linear divergence, then the QR-boundary $\partial_{*}G$ consists of exactly one point.

This result confirms the existence and triviality of the QR-boundary for groups with linear divergence, adding to known examples such as:

•

Lattices in semi-simple Lie groups of $\mathbb{Q}$ –rank $1$ and $\mathbb{R}$ –rank $\geq 2$ , uniform lattices in higher rank semi-simple Lie groups [DMS10].
•

Thompson groups $F$ , $T$ , and $V$ [GS19] and higher Thompson groups [Kod24].
•

Non-virtually cyclic groups that satisfy a law [DS05], [DMS10].
•

One-ended solvable groups [DS05], [DMS10].
•

Wreath products, permutational wreath products of groups, Houghton groups $\mathcal{H}_{m}$ with $m\geq 2$ , Baumslag-Solitar groups [I23].

Linear divergence is equivalent to wide (i.e., not having cut-points in the asymptotic cones) [DMS10], and wide groups have empty Morse boundary [DMS10]. [QR24, Question 4.4] asks if $G$ does not have an Morse element, is $P(G)$ a single point. In [GQV24], the authors answer [QR24, Question 4.4] in the affirmative when $G$ acts geometrically on a finite-dimensional CAT(0) cube complex. Our result Theorem 1.4 gives the affirmative for [QR24, Question 4.4] for the class of wide groups.

[NQ25, Theorem A, Theorem B] show that the QR poset of graph manifold groups, and more generally of Croke-Kleiner admissible groups [CK02], has QR-poset of height $2$ . This is connected to the fact that these groups have quadratic divergence. Our result shows that groups with linear divergence have QR poset of height $1$ . This naturally raises the question of whether there is a systematic relationship between divergence and QR-boundary structure.

Question 1.5.

If a group has divergence that is a polynomial of degree $d$ , is it true that its QR poset has height $d$ ?

As an application of Theorem 1.4, we establish a comprehensive result for finitely generated 3-manifold groups:

Theorem 1.6.

All finitely generated 3-manifold groups have well-defined QR-boundaries.

This result addresses cases left unresolved in [NQ25]. While [NQ25, Theorem A] showed that QR-boundaries are well-defined for fundamental groups of non-geometric 3-manifolds, the existence of QR-boundaries for geometric 3-manifolds–particularly those modeled on the Sol and Nil geometries and the broader scenario of 3-manifolds with higher genus boundaries was not completely settled.

These cases were excluded precisely because it was unknown whether their fundamental groups Sol and Nil 3-manifolds satisfy the necessary QR-assumptions. Since Sol and Nil 3-manifold groups are known to have linear divergence [Ger94], it follows from Theorem 1.4 that their fundamental groups have well-defined QR-boundary. We use this observation as a first step to conclude that all finitely generated 3-manifold groups admit a well-defined QR-boundary. Theorem 1.6 strengthens the role of QR boundaries as a tool for studying the coarse geometry of finitely generated 3-manifold groups, one of the central topics in geometric group theory.

Overview

This paper is organized as follows. Section 2 reviews preliminary concepts, including the QR-boundary construction and divergence. Section 2 proves Theorem 1.4, demonstrating that groups with linear divergence have a single-point QR-boundary. In Section 4, we give a proof of Theorem 1.6.

Acknowledgments

The author thanks Yulan Qing and Minh Nhat Doan for the helpful conversations.

2. Preliminary

2.1. Coarse geometry

In this section, we recall the construction of quasi-redirecting boundary as presented in [QR24]. Let $X$ and $Y$ be metric spaces and $f$ be a map from $X$ to $Y$ . Let ${\mathfrak{q}}=(q,Q)\in[1,\infty)\times[0,\infty)$ be a pair of constants.

Definition 2.1.

(1)

We say that $f$ is a $(q,Q)$ –quasi-isometric embedding if for all $x,y\in X$ ,

$\frac{1}{q}d(x,x^{\prime})-Q\leq d(f(x),f(x^{\prime}))\leq qd(x,x^{\prime})+Q.$
(2)

We say that $f$ is a $(q,Q)$ –quasi-isometry if it is a $(q,Q)$ –quasi-isometric embedding such that $Y=N_{Q}(f(X))$ .

Definition 2.2.

A quasi-geodesic in a metric space $X$ is a quasi-isometric embedding $\alpha:I\to X$ where $I\subset\mathbb{R}$ is a (possibly infinite) interval. That is $\alpha:I\to X$ is a $(q,Q)$ –quasi-geodesic if for all $s,t\in I$ , we have

\frac{|t-s|}{q}-Q\leq d_{X}\big{(}\alpha(s),\alpha(t)\big{)}\leq q|s-t|+Q

Remark 2.3.

We can always assume $\alpha$ is $(2q+2Q)$ –Lipschitz, and hence, $\alpha$ is continuous. By [QR24, Lemma 2.3] the Lipschitz assumption can be made without loss of generality.

Notation:

Let $o$ be a fixed base-point in $X$ . We use ${\mathfrak{q}}=(q,Q)\in[1,\infty)\times[0,\infty)$ to indicate a pair of constants. For instance, one can say $\Phi\colon X\to Y$ is a ${\mathfrak{q}}$ –quasi-isometry and $\alpha$ is a ${\mathfrak{q}}$ –quasi-geodesic ray or segment.

•

By a ${\mathfrak{q}}$ –ray we mean a ${\mathfrak{q}}$ –quasi-geodesic ray $\alpha:[0,\infty)\to X$ such that $\alpha(0)=o$ .
•

If points $x,y\in X$ on the image of $\alpha$ are given, we denote the sub-segment of $\alpha$ connecting $x$ to $y$ by $[x,y]_{\alpha}$ .
•

For $r>0$ , let $B_{r}^{\circ}\subset X$ be the open ball of radius $r$ centered at $o$ , let $B_{r}$ be the closed ball centered at $o$ and let $B_{r}^{c}=X-B_{r}^{\circ}$ . For a $\mathfrak{q}$ –ray $\alpha$ and $r>0$ , we let $t_{r}\geq 0$ denote the first time when $\alpha$ first intersects $B_{r}^{c}$ .

Lastly, if $p$ is a point on a ${\mathfrak{q}}$ –ray $\alpha$ , we use $\alpha_{[p,\infty)}$ to denote the tail of $\alpha$ starting from the point $p$ .

2.2. QR-Assumptions

In this section, we briefly review the notion QR-poset and QR-redirecting boundary from [QR24].

Definition 2.4.

Let $X$ be a geodesic metric space. Let $\alpha,\beta$ and $\gamma$ be quasi-geodesic rays in $X$ . We say

(1)

$\gamma$ eventually coincides with $\beta$ if there are times $t_{\beta},t_{\gamma}>0$ such that, for $t\geq t_{\gamma}$ , we have $\gamma(t)=\beta(t+t_{\beta})$ .
(2)

For $r>0$ , we say $\gamma$ quasi-redirects $\alpha$ to $\beta$ at radius $r$ if $\gamma|_{r}=\alpha|_{r}$ and $\beta$ eventually coincides with $\gamma$ . If $\gamma$ is a ${\mathfrak{q}}$ –ray, we say $\alpha$ can be ${\mathfrak{q}}$ –quasi-redirected to $\beta$ at radius $r$ or $\alpha$ can be ${\mathfrak{q}}$ –quasi-redirected to $\beta$ by $\gamma$ at radius $r$ . We refer to $t_{\gamma}$ as the landing time.
(3)

We say $\alpha$ is quasi-redirected to $\beta$ , denoted by $\alpha\preceq\beta$ , if there is ${\mathfrak{q}}\in[1,\infty)\times[0,\infty)$ such that for every $r>0$ , $\alpha$ can be ${\mathfrak{q}}$ –quasi-redirected to $\beta$ at radius $r$ .

Definition 2.5.

Define $\alpha\simeq\beta$ if and only if $\alpha\preceq\beta$ and $\beta\preceq\alpha$ . Then $\simeq$ is an equivalence relation on the space of all quasi-geodesic rays in $X$ .

Let $P(X)$ denote the set of all equivalence classes of quasi-geodesic rays under $\simeq$ . For a quasi-geodesic ray $\alpha$ , let $[\alpha]\in P(X)$ denote the equivalence class containing $\alpha$ . We extend $\preceq$ to $P(X)$ by defining $[\alpha]\preceq[\beta]$ if $\alpha\preceq\beta$ . Note that this does not depend on the chosen representative in the given class. The relation $\preceq$ is a partial order on elements of $P(X)$ . We call $P(X)$ the QR-poset of $X$ .

QR-Assumption 0: (No dead ends)

The metric space $X$ is proper and geodesic. Furthermore, there exists a pair of constants ${\mathfrak{q}}_{0}$ such that every point $x\in X$ lies on an infinite ${\mathfrak{q}}_{0}$ –quasi-geodesic ray.

QR-Assumption 1: (Quasi-geodesic representative)

For ${\mathfrak{q}}_{0}$ as in QR-Assumption 0, every equivalence class of quasi-geodesics $\mathbf{a}\in P(X)$ contains a ${\mathfrak{q}}_{0}$ –ray. We fix such a ${\mathfrak{q}}_{0}$ –ray, denote it by $\underline{a}\in\mathbf{a}$ , and call it a central element of $\mathbf{a}$ .

QR-Assumption 2: (Uniform redirecting function)

For every $\mathbf{a}\in P(X)$ , there is a function

f_{\mathbf{a}}:\,[1,\infty)\times[0,\infty)\to[1,\infty)\times[0,\infty),

called the redirecting function of the class $\mathbf{a}$ , such that if $\mathbf{b}\prec\mathbf{a}$ then any ${\mathfrak{q}}$ –ray $\beta\in\mathbf{b}$ can be $f_{\mathbf{a}}({\mathfrak{q}})$ –quasi-redirected to $\underline{a}$ .

Quasi-redirecting boundary (QR-boundary):

Once a proper geodesic metric space $X$ satisfies all three QR-Assumptions, there is a “cone-line” topology on the poset $P(X)$ described on [QR24]. This poset $P(X)$ , when equipped with this topology, is called the quasi-redirecting boundary (QR boundary) of $X$ and denoted by $\partial_{*}X$ . Since we don’t use this topology on $P(X)$ in an essential way in this paper, we refer the reader to [QR24] for the detailed discussion.

A remarkable fact about QR-boundary is the following result.

Theorem 2.6 ([QR24, Theorem B, Theorem C]).

Let $X,Y$ be proper geodesic metric spaces satisfying all three QR-Assumptions.

(1)

A quasi-isometry $f\colon X\to Y$ induces a homeomorphism between $\partial_{*}X$ and $\partial_{*}Y$ .
(2)

Sublinearly Morse boundaries are topological subspaces of $\partial_{*}X$ .

2.3. Divergence of groups

In this section, we briefly review the definition of divergence from [DMS10].

Definition 2.7.

Let $\mathcal{F}$ be the collection of all functions from positive reals to positive reals. Let $f$ and $g$ be arbitrary elements of $\mathcal{F}$ . The function $f$ is dominated by a function $g$ , denoted by $f\preceq g$ , if there are positive constants $A$ , $B$ , $C$ , $D$ and $E$ such that

f(x)\leq A\,g(Bx+C)+Dx+E\quad\text{for all $x$.}

Two functions $f$ and $g$ are equivalent, denoted by $f\sim g$ , if $f\preceq g$ and $g\preceq f$ .

Remark 2.8.

The relation $\sim$ is an equivalence relation on the set $\mathcal{F}$ . Let $f$ and $g$ be two polynomial functions with degree at least $1$ in $\mathcal{F}$ , then it is not hard to show that they are equivalent if and only if they have the same degree. Moreover, all exponential functions of the form $a^{bx+c}$ , where $a>1$ , $b>0$ are equivalent.

Since we mainly work on Cayley graphs of finitely generated groups, we assume in this section that our metric spaces are geodesic, proper and periodic spaces.

For such a space $X$ , given three points $a,b,c\in X$ and parameters $\delta\in(0,1)$ and $\gamma\geq 0$ we define divergence $\operatorname{div}_{\gamma}(a,b,c;\delta)$ to be the infimum of lengths of paths that connect $a$ to $b$ outside $B^{o}(c,\delta r-\gamma)$ , the open ball around $c$ of radius $\delta r-\gamma$ , if this exists. We define it to be infinite otherwise. Here $r=\operatorname{min}\{d(c,a),d(c,b)\}$ . We then define

\operatorname{Div}(n,\delta)=\operatorname{sup}_{a,b,c\in X,d(a,b)\leq n}\operatorname{div}(a,b,c;\delta)

Since a space has more than one end then its divergence is infinite, we thus restrict to one-ended spaces. Furthermore, we can fix a third point $c=x_{0}$ in the definition and assume that $a,b$ are in the sphere $S_{r}:=S(x_{0},r)$ and $\operatorname{Div}_{\gamma}(n,\delta)$ can be modified to:

\operatorname{Div}_{\gamma}(n,\delta)=\operatorname{sup}_{a,b\in S(x_{0},r)}\operatorname{div}_{\gamma}(a,b,x_{0};\delta)

It is shown by [DMS10] that $\operatorname{Div}_{\gamma}(n,\delta)$ is independent of $\gamma$ and $\delta$ up to $\sim$ for any $\delta\leq 1/2$ and $\gamma\geq 2$ and is invariant under quasi-isometry up to $\sim$ . Thus in this paper, we think of $\operatorname{Div}(X)$ as a function of $n$ , defining it to be equal to $\operatorname{Div}_{2}(n,1/2)$ . We say that the divergence is linear if $\operatorname{Div}_{2}(n,1/2)\sim n$ , quadratic if $\operatorname{Div}_{2}(n,1/2)\sim n^{2}$ , and so on.

3. Quasi-redirecting boundaries of groups with linear divergence

In this section, we are going to prove Theorem 1.4. To prove Theorem 1.4, we show that the cayley graph $X$ of $G$ (with respect to a finite generating set), any two quasi-geodesic rays $\alpha$ and $\beta$ are equivalent under the QR-relation. We achieve this by constructing quasi-geodesic rays that direct $\alpha$ to $\beta$ by using the linear divergence property.

We need the following lemmas.

Lemma 3.1 ([QR24, Lemma 2.6]).

Let $X$ be a metric space that satisfies QR-Assumption 0. (Nearest-point projection surgery) Consider a point $x\in X$ and a $(q,Q)$ –quasi-geodesic segment $\beta$ connecting a point $z\in X$ to a point $w\in X$ . Let $y$ be a closest point in $\beta$ to $x$ . Then

\gamma=[x,y]\cup[y,z]_{\beta}

is a $(3q,Q)$ –quasi-geodesic.

Lemma 3.2.

[NQ25, Lemma 2.9] Let $\alpha,\beta$ be quasi-geodesic rays. Suppose there exists constants $\mathfrak{q}$ and a sequence of points $\{x_{n}\}$ on $\alpha$ such that $\operatorname{norm}{x_{n}}\to\infty$ and the following holds. For every $n$ , there exists a $\mathfrak{q}$ -ray $\gamma_{n}$ such that $\gamma_{n}$ eventually coincides with $\beta$ , and $\gamma_{n}$ and $\alpha$ are identical on the subsegment $[o,x_{n}]_{\alpha}$ . Then $\alpha$ can be $\mathfrak{q}$ -quasi-redirected to $\beta$ .

The following lemma follows from the proof of [QR24, Lemma 3.5].

Lemma 3.3.

[QR24, Lemma 3.5] Let $X$ be a proper, geodesic, metric space and let $\alpha$ be a $\mathfrak{q}$ –ray. Then there exists a geodesic ray $\bar{\alpha}$ such that $\bar{\alpha}$ is $(3q,Q)$ –quasi-redirected to $\alpha$ .

Lemma 3.4.

[Tra19, Lemma 3.3]] For each $C>1$ and $\rho\in(0,1]$ there is a constant $L=L(C,\rho)\geq 1$ such that the following holds. Let $r$ be an arbitrary positive number and $\gamma$ a path with the length $\ell(\gamma)<Cr$ and $d(x,y)>r$ . Then there is an $(L,0)$ –quasi-geodesic $\alpha$ connecting two points $x$ , $y$ such that the image of $\alpha$ lies in the $\rho r$ –neighborhood of $\gamma$ and $\ell(\alpha)<\ell(\gamma)$ .

Lemma 3.5 (Annulus Surgery).

Let $X$ be a proper geodesic metric space. Given $\delta>\epsilon>0$ and a constant $C>0$ . Given two pairs of constants $(q_{1},Q1),(q_{2},Q_{2})\in[1,\infty)\times[0,\infty)$ . Then there exists a constant $M=M(\delta,\epsilon,q_{1},Q_{1},q_{2},Q_{2},C)$ such that the following holds.

Let $\alpha$ be a $(q_{1},Q_{1})$ –quasi-geodesic based at $o$ with $\alpha_{+}$ in the sphere $S_{\epsilon r}:=S(o,\epsilon r)$ and lies entirely in the ball $B_{\epsilon r}$ with $r>1$ . Let $\zeta$ be a geodesic ray based at $o$ and passes through $\alpha_{+}$ . Let $p$ denote the intersection point of $\zeta$ with the sphere $S_{\delta r}$ . Let $\gamma$ be a $(q_{2},Q_{2})$ –quasi-geodesic based at $p$ such that $\beta$ lies entirely outside the open ball $B^{o}_{\delta r}$ and $\ell(\beta)\leq Cr$ . Then the concatenation $\sigma:=\alpha\cup[\alpha_{+},p]\cup\beta$ is a $(M,M)$ –quasi-geodesic.

Refer to caption — Figure 1. The cacatenation $\alpha\cup[\alpha_{+},p]\cup\gamma$ is a quasi-geodesic.

Proof.

To see this, for every $x\neq y\in\sigma$ , we are going to show that the ratio

\frac{\ell([x,y]_{\sigma})}{d(x,y)}

is bounded above by a uniform constant.

According to Lemma 3.1, the concatenations

\alpha\cup[\alpha_{+},p],[\alpha_{+},p]\cup\gamma

are $(3q_{1},Q_{1})$ –quasi-geodesic and $(3q_{2},Q_{2})$ –quasi-geodesic respectively.

We thus only need to consider the case $x\in\alpha$ and $y\in\gamma$ . On a one hand, we have

	$\displaystyle\ell([x,y]_{\sigma})$	$\displaystyle\leq\ell(\alpha)+d(\alpha_{+},p)+\ell(\gamma)$
		$\displaystyle\leq q_{1}d(o,\alpha_{+})+Q_{1}+(\delta-\epsilon)r+Cr$
		$\displaystyle\leq(q_{1}+\delta-\epsilon+C)r+Q_{1}<(q_{1}+\delta-\epsilon+C+Q_{1})r$

On the other hand, since $x,y$ lie outside the annulus $B_{\delta r}\backslash B^{o}_{\epsilon r}$ , we have

d(x,y)\geq(\delta-\epsilon)r

Thus we have

\frac{\ell([x,y]_{\sigma})}{d(x,y)}\leq\frac{(q_{1}+Q_{1}+C+\delta-\epsilon)r}{(\delta-\epsilon)r}=\frac{q_{1}+Q_{1}+C+\delta-\epsilon}{\delta-\epsilon}

Combining with cases $x,y\in\alpha\cup[\alpha_{+},p]$ , $x,y\in[\alpha_{+},p]\cup\gamma]$ (which are $(3q_{1},Q_{1})$ and $(3q_{2},Q_{2})$ –quasi-geodesics respectively), there is a constant $M$ depending only on constants $q_{1},Q_{1},q_{2},Q_{2},\delta,\epsilon,C$ so that $\sigma$ is a $(M,M)$ –quasi-geodesic. ∎

The following lemma is a slight modification of [QRT22, Lemma 4.3]. We include it here for completeness.

Lemma 3.6.

(Quasi-geodesic ray to geodesic ray surgery) Let $\epsilon\in(0,1)$ . Let $\beta$ be a geodesic ray and $\gamma$ be a $(q,Q)$ –ray. Suppose that there exists an increasing sequence $\{r_{n}\}$ such that for every $n$ , $d(\beta(r_{n}),\gamma)\leq\epsilon r_{n}$ . Then $\gamma$ can be $(9q,Q)$ –quasi-redirected to the geodesic ray $\beta$ .

Proof.

Let $q_{n}$ be a point in $\gamma$ that is closest to $\beta(r_{n})$ and let $R_{n}>0$ be such that the ball of radius $R_{n}$ centered at $o$ contains $[o,q_{n}]_{\gamma}$ . Now let $q^{\prime}_{n}$ be the point in $[o,q_{n}]_{\gamma}$ closest to $\beta(R_{n})$ . Then

	$\displaystyle d(o,q^{\prime}_{n})$	$\displaystyle\geq d(o,\beta(R_{n}))-d(\beta(R_{n}),q^{\prime}_{n})=R_{n}-d(\beta(R_{n}),q_{n})$
		$\displaystyle\geq R_{n}-(d(\beta(R_{n}),\beta(r_{n}))+d(\beta(r_{n}),q_{n}))$
		$\displaystyle\geq R_{n}-(R_{n}-r_{n})-\epsilon r_{n}=(1-\epsilon)r_{n}$

By Lemma 3.1, the concatenation $\zeta:=[o,q^{\prime}_{n}]_{\gamma}\cup[q^{\prime}_{n},\beta(R_{n})]$ is a $(3q,Q)$ –quasi-geodesic. Furthermore, $d(o,q^{\prime}_{n})\leq R$ , it follows that the projection of any point on the geodesic $\beta([R_{n},\infty))$ to $\zeta$ is the point $\beta(R_{n})$ . By Lemma 3.1 again, the concatenation $\xi:=\zeta\cup\beta([R_{n},\infty)$ is a $(9q,Q)$ –quasi-geodesic. Since $d(o,q^{\prime}_{n})\geq(1-\epsilon)r_{n}$ , it follows that $\xi$ is identical with $\gamma$ in the open ball $B^{o}_{(1-\epsilon)r_{n}}$ . In other words, $\gamma$ can be $(9q,Q)$ –quasi-redirected to the geodesic ray $\beta$ at the radius $r^{\prime}_{n}:=(1-\epsilon)r_{n}$ . As $r^{\prime}_{n}\to\infty$ , it follows from Lemma 3.2 that $\gamma$ can be $(9q,Q)$ –quasi-redirected to $\beta$ . The lemma is proved. ∎

We are now ready for the proof of Theorem 1.4.

Proof of Theorem 1.4.

Fix a finite generating set for $G$ , and let $X$ the Cayley graph of $G$ with respect to this generating set. We aim to prove that the poset $P(X)$ consists of exactly one point and satisfies all three QR-Assumptions.

Consider two $\mathfrak{q}$ -rays $\alpha$ and $\beta$ in $X$ , both based at a vertex $o$ . By Lemma 3.3, there exist geodesic rays $\bar{\alpha}$ and $\bar{\beta}$ in $X$ such that $\bar{\alpha}\preceq\alpha$ and $\bar{\beta}\preceq\beta$ . We will show that $\alpha\preceq\bar{\beta}$ , which implies $\alpha\preceq\beta$ . By a symmetric argument, we can also establish $\beta\preceq\alpha$ , thus proving $\alpha\sim\beta$ (and hence the poset $P(X)$ consists of exactly one point).

Since the divergence of $X$ is linear, it follows that there exists a constant $C>0$ such that for every $r>1$ then

\operatorname{Div}_{2}(r,1/2)\leq Cr

That is for every $x\neq y$ in the sphere $S(o,r)$ , there exists a path $\alpha$ connecting $x$ to $y$ lying outside the open ball $B^{o}_{\frac{r}{2}-2}$ such that $\ell(\alpha)\leq Cr$ .

Recall that $t_{r}$ denote the first time the path $\alpha$ first intersects $B^{c}_{r}:=X\backslash B^{o}_{r}$ . Since $\bar{\beta}$ is a geodesic ray, we thus have $\bar{\beta}(t_{r})=\bar{\beta}(r)$ .

We consider the following cases.

Case 1: There exists a strictly increasing sequence $(r_{n})_{n\in\mathbb{N}}$ such that for each $n\in\mathbb{N}$ , the inequality

d(\bar{\beta}(r_{n}),\alpha(t_{\frac{35r_{n}}{72}}))\leq\frac{39r_{n}}{72}

holds. It follows that

d(\beta(r_{n}),\alpha)\leq\frac{39r_{n}}{72}

By Lemma 3.6, $\alpha$ can be $(9q,Q)$ –quasi-redirected to $\bar{\beta}$ .

Case 2: Case 1 does not hold, that is, for every increasing sequence $r_{1}<r_{2}<\ldots$ , there exists $n_{0}\in\mathbb{N}$ such that

d(\bar{\beta}(r_{n_{0}}),\alpha(t_{\frac{35r_{n_{0}}}{72}}))>\frac{39r_{n_{0}}}{72}

It follows that there exists an increasing subsequence $4<s_{1}<s_{2}<\ldots$ of $\{r_{n}\}$ such that

d(\bar{\beta}(s),\alpha(t_{\frac{35s}{72}}))>\frac{39s}{72}

for all $s=s_{i}$ .

Let $\zeta$ be a geodesic ray based at $o$ and passes through $p^{\prime}:=\alpha(t_{\frac{35s}{72}})$ . Let $p$ denote the intersection point of $\zeta$ with $S(o,s)$ . We have

d(\bar{\beta}(s),p)\geq d(\bar{\beta}(s),p^{\prime})-d(p^{\prime},p)\geq\frac{39s}{72}-(s-\frac{35s}{72})=\frac{s}{36}

Claim 1: There exists a constant $L=L(C)$ , and an $(L,0)$ –quasi-geodesic $\sigma$ connecting $p$ to $\bar{\beta}(s)$ such that $\ell(\sigma)<Cs$ and $\sigma$ lies in the annulus $B_{(1+C)s}\backslash B^{o}_{\frac{71s}{144}-2}$ .

Indeed, since both points $\bar{\beta}(s)$ and $p$ lie in the sphere $S(o,s)$ and the divergence of $X$ is linear, it follows that there is a path $\tau$ connecting $p$ to $\bar{\beta}(s)$ lying outside the open ball $B^{o}(o,s/2-2)$ and the length of $\tau$ satisfies

\ell(\tau)\leq Cs

By Lemma 3.4, there exists a constant $L$ depending only on $C$ , an $(L,0)$ –quasi-geodesic $\sigma$ from $p$ to $\bar{\beta}(s)$ so that $\ell(\sigma)<\ell(\tau)<Cs$ and $\sigma\subset\mathcal{N}_{\frac{s}{144}}(\tau)$ where $\mathcal{N}_{\frac{s}{144}}(\tau)$ we mean the $\frac{s}{144}$ –neighborhood of $\tau$ . Since $\tau$ lies outside the ball $B(o,s/2-2)$ , it follows that $\sigma$ lies outside the ball $B_{\frac{71s}{144}-2}$ .

Let $R:=\operatorname{max}\{d(x,o):x\in\sigma\}$ .Then we have $\sigma\subset B_{R}$ . Note that $R\leq\ell(\sigma)+d(\bar{\beta}(s),o)\leq Cs+s=(1+C)s$ . Thus $\sigma$ lies inside the ball $B_{(1+C)s}$ .

In the rest of the proof, we will construct the desired quasi-geodesic.

Let $a$ be the intersection point of the geodesic ray $\zeta$ with the sphere $S_{\frac{71s}{144}-2}$ . Let $b\in\sigma$ be the nearest point to $a$ . Since $\sigma$ is an $(L,0)$ –quasi-geodesic, it follows that the concatenation $\gamma_{1}:=[a,b]\cup[b,\bar{\beta}(s)]_{\sigma}$ is a $(3L,0)$ –quasi-geodesic by Lemma 3.1.

Let $e:=\bar{\beta}((1+C)s)$ , and let $d\in\gamma_{1}=[a,b]\cup[b,\bar{\beta}(s)]_{\sigma}$ be the nearest point to $e$ (we refer the reader to Figure 2). Again by Lemma 3.1, we have that the concatenation $\gamma_{2}:=[e,d]\cup[d,a]_{\gamma_{1}}$ is a $(9L,0)$ –quasi-geodsic.

We have

	$\displaystyle\ell(\gamma_{2})$	$\displaystyle\leq d(e,d)+d(a,b)+\ell(\sigma)$
		$\displaystyle\leq d(e,\bar{\beta}(s))+d(a,p)+Cs$
		$\displaystyle\leq Cs+2+\frac{74s}{144}+Cs$
		$\displaystyle\leq(\frac{75}{144}+1+2C)s$

Applying Lemma 3.5, we have that the concatenation

\xi:=[o,p^{\prime}]_{\alpha}\cup[p^{\prime},a]\cup\gamma_{2}

is a $(M,M)$ –quasi-geodesic where $M$ is the constant given by Lemma 3.5.

Now, every point $u\in\bar{\beta}|_{(1+C)s}$ , its nearest point projection on $\xi$ is $e$ since $\xi$ lies entirely in the ball $B_{(1+C)s}$ . According to Lemma 3.1, the concatenation $\xi\cup\bar{\beta}|_{\geq(1+C)s}$ is a $(3M,M)$ –quasi-geodesic.

Thus, we can redirect $\alpha$ to the geodesic ray $\bar{\beta}$ at radius $s$ . Since this can be done for an increasing sequence of radius $s$ (i.e, the sequence $s_{1}<s_{2}<\ldots$ ), it follows from Lemma 3.2 that $\alpha$ can be $(3M,M)$ –quasi-redirected to $\bar{\beta}$ .

In conclusion, enlarging constants $M$ if necessary, we have that $\alpha$ can be $(3M,M)$ –quasi-redirected to $\bar{\beta}$ in both Case 1 and Case 2.

Since $\bar{\beta}$ can be $(3q,Q)$ –quasi-redirected to $\beta$ by Lemma 3.3, it follows from [QR24, Lemma 3.2] that $\alpha$ can be $(3q+3M+1,Q+M)$ –quasi-redirected to $\beta$ .

Using symmetric argument, it can be shown that $\beta$ is $(3q+3M+1,Q+M)$ –quasi-redirected to $\alpha$ . Therefore

\alpha\sim\beta

In particular, the poset $P(X)$ consists of exactly one point and satisfies all three QR-Assumptions. Therefore $\partial_{*}X$ is well-defined and consists of only one point.

∎

4. Quasi-redirecting boundary of finitely generated 3-manifold groups

In this section, we are going to prove Theorem 1.6 which says all finitely generated 3-manifold groups have well-defined QR-boundaries.

A compact orientable irreducible $3$ –manifold $M$ with empty or tori boundary is called geometric if its interior admits a geometric structure in the sense of Thurston which are $3$ –sphere, Euclidean $3$ –space, hyperbolic $3$ –space, $S^{2}\times\mathbb{R}$ , $\mathbb{H}^{2}\times\mathbb{R}$ , $\widetilde{SL}(2,\mathbb{R})$ , Nil and Sol. Otherwise, it is called non-geometric. The QR-boundaries have been studied in [NQ25].

Theorem 4.1.

[NQ25, Theorem A] Let $M$ be an non-geometric 3-manifold. Then the fundamental group $G=\pi_{1}(M)$ satisfies the QR-Assumptions and hence $\partial_{*}G$ is well-defined.

We now address the geometric case.

Lemma 4.2 ( Geometric case).

Let $M$ be an geometric 3-manifold. Then $\partial_{*}\pi_{1}(M)$ is well-defined.

Proof.

$M$ has a geometric structure modeled on eight geometries in the sense of Thurston: $S^{3}$ , $\mathbb{R}^{3}$ , $S^{2}\times\mathbb{R}$ , Nil, $\widetilde{SL(2,\mathbb{R})}$ , $\mathbb{H}^{2}\times\mathbb{R}$ , $\mathbb{H}^{3}$ , and Sol.

(1)

If the geometry of $M$ is spherical, then its fundamental group is finite, and hence $\partial_{*}(\pi_{1}(M))$ is empty.
(2)

If the geometry of $M$ is $\mathbb{E}^{3}$ , then there exists a finite index subgroup $K\leq\pi_{1}(M)$ such that $K$ is isomorphic to $\mathbb{Z}^{3}$ . It follows that $\pi_{1}(M)$ is quasi-isometric to $\mathbb{E}^{3}$ , and hence $\partial_{*}(\pi_{1}(M))$ consists of only one point as $\mathbb{E}^{3}$ has linear divergence.
(3)

If the geometry of $M$ is $S^{2}\times\mathbb{R}$ , then there exists a finite index subgroup $K\leq\pi_{1}(M)$ such that $K$ is isomorphic to $\mathbb{Z}$ . It follows that $\pi_{1}(M)$ is quasi-isometric to $\mathbb{Z}$ , and hence $\partial_{*}(\pi_{1}(M))$ is well-defined and consists of only two points.
(4)

If the geometry of $M$ is $\mathbb{H}^{2}\times\mathbb{R}$ , then $M$ is finitely covered by $M^{\prime}=\Sigma\times S^{1}$ where $\Sigma$ is a compact surface with negative Euler characteristic. It follows that $\pi_{1}(M)$ is quasi-isometric to the direct product $\pi_{1}(\Sigma)\times\mathbb{Z}$ , and hence $\partial_{*}(\pi_{1}(M))$ is well-defined and consists of only one point.
(5)

If $M$ has a geometry modeled on $\widetilde{SL(2,\mathbb{R})}$ , then $\partial_{*}(\pi_{1}(M))$ is well-defined and consists of only one point since two geometries $\mathbb{H}^{2}\times\mathbb{R}$ and $\widetilde{SL(2,\mathbb{R})}$ are quasi-isometric and QR-boundary is a quasi-isometric invariant (Theorem 2.6).
(6)

If $M$ has a geometry modeled on $\mathbb{H}^{3}$ then $M$ is a hyperbolic 3-manifold with finite volume. $\partial_{*}(\pi_{1}(M))$ is well-defined as shown in [QR24].
(7)

Finally, If $M$ has a geometry modeled on Sol or Nil then $\pi_{1}(M)$ has linear divergence (see the first and second paragraphs in the proof of [Ger94, Theorem 4], and thus Theorem 1.4 implies that $\partial_{*}(\pi_{1}(M))$ is well-defined and consists of only one point.

∎

Below, we explain how one might reduce the study of all finitely generated 3–manifold groups to the case of compact, orientable, irreducible, $\partial$ –irreducible 3–manifold groups.

Let $M$ be 3-manifold with finitely generated fundamental group. By Scott’s Core Theorem, $M$ contains a compact codimension zero submanifold whose inclusion map is a homotopy equivalence [Sco73], and thus also an isomorphism on fundamental groups. We thus can assume $M$ is compact. Since QR-boundary is a quasi-isometric invariant (Theorem 2.6), we can assume that M is orientable by passing to a double cover if necessary.

We can also assume that $M$ is irreducible and $\partial$ –irreducible by the following reason: Since $M$ is a compact, orientable $3$ –manifold, it decomposes into irreducible, $\partial$ –irreducible pieces $M_{1},\dots,M_{k}$ (by the sphere-disc decomposition). In particular, $\pi_{1}(M)$ is the free product

\pi_{1}(M)=\pi_{1}(M_{1})*\pi_{1}(M_{2})*\cdots*\pi_{1}(M_{k})

Let $G_{i}=\pi_{1}(M_{i})$ . We remark here that $\pi_{1}(M)$ is hyperbolic relative to the collection $\mathbb{P}=\{G_{1},\cdots,G_{k}\}$ . According to [NQ25, Theorem D], if the QR-boundary exist for each peripheral subgroup $G_{i}$ then the QR-boundary of $\pi_{1}(M)$ exists. In other words, for the purpose of showing QR-boundary exists for $\pi_{1}(M)$ , we only need to focus on the case where the manifold $M$ is compact, connected, orientable, irreducible, and $\partial$ –irreducible.

If $M$ has empty or toroidal boundary, the existence of QR-boundary of $\pi_{1}(M)$ would follow from Theorem 4.1 and Lemma 4.2.

We note that the compact, connected, orientable, irreducible and $\partial$ –irreducible manifold $M$ could have boundary components that are higher genus surfaces. The following lemma addresses certain manifolds with higher genus boundary.

Lemma 4.3.

Let $M$ be a compact, orientable, irreducible, $\partial$ –irreducible $3$ –manifold which has at least one boundary component of genus at least $2$ . Then the fundamental group $G=\pi_{1}(M)$ satisfies the QR-Assumptions and hence $\partial_{*}G$ is well-defined.

Proof.

As in [Sun20, Section 6.3], we can paste compact hyperbolic 3-manifolds with totally geodesic boundaries to the higher genus boundary components of $M$ to obtain a finite volume hyperbolic manifold $N$ (in case $M$ has trivial torus decomposition) or a mixed 3-manifold (in case $M$ has non-trivial torus decomposition). By [NQ25, Theorem A], $\pi_{1}(N)$ has well-defined QR-boundary. The new manifold $N$ satisfies the following properties.

(1)

$M$ is a submanifold of $N$ with incompressible tori boundary.
(2)

The torus decomposition of $M$ also gives the torus decomposition of $N$ .
(3)

Each piece of $M$ with a boundary component of genus at least $2$ is contained in a hyperbolic piece of $N$ .

Since $N$ contains at least one hyperbolic piece, we equip $N$ with a nonpositively curved metric as in [Leeb95]. This metric induces a metric on the universal cover $\tilde{N}$ .

Let $M^{\prime}_{1},\dots,M^{\prime}_{k}$ be the pieces of $M$ that satisfies (3). Let $N^{\prime}_{i}$ be the hyperbolic piece of $N$ such that $M^{\prime}_{i}$ is contained in $N^{\prime}_{i}$ . We remark here that it has been proved in [NS19] that the inclusion of the subgroup $\pi_{1}(M)$ in $\pi_{1}(N)$ is a quasi-isometric embedding (see the proof of Case 1.2 in the proof of Theorem 1.3 in [NS19]), and hence the inclusion $\widetilde{M}\to\widetilde{N}$ is a $(\lambda_{1},c_{1})$ –quasi-isometric embedding for some uniform constants $\lambda_{1},c_{1}$ .

Since QR-boundary is a quasi-isometric invariant (see Theorem 2.6), it suffices to show that the QR-boundary exists for the universal cover $\widetilde{M}$ .

Fix a base point $o\in\widetilde{M}$ . Let $\alpha$ and $\beta$ be two $\mathfrak{q}$ –rays in $\widetilde{M}$ based at $o$ . Since the inclusion $\widetilde{M}\to\widetilde{N}$ is a $(\lambda_{1},c_{1})$ –quasi-isometric embedding, $\alpha$ and $\beta$ are $\mathfrak{q}^{\prime}$ –rays in $\widetilde{N}$ for some $q^{\prime}=q^{\prime}(q,Q,\lambda_{1},c_{1})$ and $Q^{\prime}=Q^{\prime}(q,Q,\lambda_{1},c_{1})$ .

Claim 1: There exists a function $\mathcal{M}\colon[1,\infty)\times[0,\infty)\to[1,\infty)\times[0,\infty)$ such that the following holds. At every radius $r$ , if $\alpha$ can be quasi-redirected to $\beta$ at radius $r$ via a $(A,B)$ –quasi-geodesic $\gamma$ in $\widetilde{N}$ then $\alpha$ can be quasi-redirected to $\beta$ at radius $r$ via a $\mathcal{M}(A,B)$ –quasi-geodesic $\gamma^{\prime}$ which lies entirely in $\widetilde{M}\subset\widetilde{N}$ .

Proof of the claim.

Indeed, let $s$ be the landing time of $\gamma$ on $\beta$ . By the construction of the manifold $N$ , the restriction of $\gamma$ on $[0,s]$ can be decomposed into a concatenation

\gamma|_{[0,s]}=\alpha_{1}\cdot\beta_{1}\cdot\alpha_{2}\cdot\beta_{2}\cdots\alpha_{\ell}\cdot\beta_{\ell}\cdot\alpha_{\ell+1}

such that:

•

For each $j$ , the subpath $\alpha_{j}$ is a subset of $\tilde{M}$ , and $\beta_{j}$ intersects $\tilde{M}$ only at its endpoints. Here $\alpha_{1}$ and $\alpha_{\ell+1}$ might degenerate to points.
•

Moreover, there are pieces $\tilde{M}^{\prime}_{j}$ and $\tilde{N}^{\prime}_{j}$ of $\tilde{M}$ and $\tilde{N}$ respectively such that $\tilde{M}^{\prime}_{j}\subset\tilde{N}^{\prime}_{j}$ , $\beta_{j}\subset\tilde{N}^{\prime}_{j}$ , and the endpoints of $\beta_{j}$ lies in $\widetilde{\Sigma}_{j}\subset\tilde{M}^{\prime}_{j}$ where $\widetilde{\Sigma}_{j}$ is the universal cover of some boundary component of genus at least 2 of $M_{j}$ .

Since each inclusion $\pi_{1}(M_{j})\to\pi_{1}(N_{j})$ is a quasi-isometric embedding, and $\pi_{1}(N_{j})$ is a hyperbolic group, the subgroup $\pi_{1}(M_{j})$ is a quasi-convex in $\pi_{1}(N_{j})$ by Morse Lemma.

Since there are finitely many pieces $M_{j}$ ’s, it follows that there exists a function $\mathcal{M}^{\prime}\colon[1,\infty)\times[0,\infty)\to[1,\infty)\times[0,\infty)$ such that for every $(\lambda,c)$ –quasi-geodesic $\beta$ in $\widetilde{N}_{j}$ with endpoints in $\widetilde{M}_{j}$ will eventually lies in the $\mathcal{M}^{\prime}(\lambda,c)$ –distance from a geodesic in $\widetilde{M}_{j}$ connecting the two endpoints of the quasi-geodesic $\beta$ . Thus each $(A,B)$ –quasi-geodesic $\beta_{j}$ in $\widetilde{N}_{j}$ lies within a $R$ -distance from a geodesic in $\widetilde{M}_{j}$ connecting two endpoints of $\beta_{j}$ , denoted $\beta^{\prime}_{j}$ in $\widetilde{M}_{j}$ . Define

\bar{\gamma}:=\alpha_{1}\cdot\beta^{\prime}_{1}\cdot\alpha_{2}\cdot\beta^{\prime}_{2}\cdots\alpha_{\ell}\cdot\beta^{\prime}_{\ell}\cdot\alpha_{\ell+1}

and

\gamma^{\prime}:=\bar{\gamma}\cup\gamma|_{[s,\infty)}

that lies entirely in $\widetilde{M}$ . Since each $\beta_{j}$ lies within a $R$ –distance from $\beta^{\prime}_{j}$ and $\alpha_{j}\subset\gamma$ and $\gamma$ is a quasi-geodesic ray, it follows that $\gamma^{\prime}$ is a $\mathcal{M}(A,B))$ –quasi-geodesic ray. ∎

Claim 2: $\widetilde{M}$ satisfies all three QR-Assumptions, and hence it has well-defined QR-boundary.

For every quasi-geodesic ray $\alpha$ in $\widetilde{M}$ , it is also a quasi-geodesic ray in $\widetilde{N}$ since the inclusion $\widetilde{M}\to\widetilde{N}$ is a quasi-isometric embedding. In [NQ25], the class $[\alpha]\in P(\widetilde{N})$ contains a geodesic representative $\underline{\alpha}$ and this geodesic lies in $\widetilde{M}$ by our construction of $N$ (i.e, the torus decomposition of $M$ also gives the torus decomposition of $N$ ) . We thus consider $\underline{\alpha}$ is a representative of $[a]\in P(\widetilde{M})$ as well.

Given $\mathbf{a}\in P(\widetilde{M})$ , we consider $i(\mathbf{a})\in P(\widetilde{N})$ where $i\colon P(\widetilde{M})\to P(\widetilde{N})$ is the inclusion. By [NQ25, Theorem A] there is a function

f_{i(\mathbf{a})}:\,[1,\infty)\times[0,\infty)\to[1,\infty)\times[0,\infty),

called the redirecting function of the class $i(\mathbf{a})$ . Now let $\mathbf{b}\in P(\widetilde{M})$ such that $\mathbf{b}\prec\mathbf{a}$ . In particular, $\mathbf{b}\prec i(\mathbf{a})$ such that if $\mathbf{b}\prec i(\mathbf{a})$ where $\mathbf{b}\in P(\widetilde{N})$ then any ${\mathfrak{q}}$ –ray $\beta\in\mathbf{b}$ can be $f_{i(\mathbf{a})}({\mathfrak{q}})$ –quasi-redirected to $\underline{a}$ via $f_{i(\mathbf{a})}(\mathfrak{q})$ –rays in $\widetilde{N}$ . According to Claim 1, such $f_{i(\mathbf{a})}(\mathfrak{q})$ –rays in $\widetilde{N}$ can be modified to be $\mathcal{M}(f_{i(\mathbf{a})}(\mathfrak{q}))$ –rays in $\widetilde{M}$ that quasi-redirect $\beta$ to $\underline{a}$ . This shows that $\widetilde{M}$ would satisfy all three QR-Assumptions and thus $\pi_{1}(M)$ has well-defined QR-boundary. ∎

Proof of Theorem 1.6.

By applying Scott’s Core Theorem and, if needed, passing to a double cover, combined with the quasi-isometric invariance of the QR-boundary, we may assume $M$ is compact and orientable. We then employ sphere-disc decomposition to decompose $M$ into irreducible, $\partial$ -irreducible manifolds $M_{1},M_{2},\ldots,M_{k}$ . Consequently, the fundamental group decomposes as a free product: $\pi_{1}(M)=\pi_{1}(M_{1})*\pi_{1}(M_{2})*\cdots*\pi_{1}(M_{k})$ . By [NQ25, Theorem D], the existence of a QR-boundary for each $\pi_{1}(M_{i})$ implies its existence for $\pi_{1}(M)$ . This follows from Theorem 4.1, Lemma 4.2, and Lemma 4.3, completing the proof. ∎

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