Quasiconvexity in 3-manifold groups

Abstract

In this paper, we study strongly quasiconvex subgroups in a finitely generated 3-manifold group $\pi _1(M)$. We prove that if M is a compact, orientable 3-manifold that does not have a summand supporting the Sol geometry in its sphere-disc decomposition then a finitely generated subgroup $H \le \pi _1(M)$ has finite height if and only if H is strongly quasiconvex. On the other hand, if M has a summand supporting the Sol geometry in its sphere-disc decomposition then $\pi _1(M)$ contains finitely generated, finite height subgroups which are not strongly quasiconvex. We also characterize strongly quasiconvex subgroups of graph manifold groups by using their finite height, their Morse elements, and their actions on the Bass-Serre tree of $\pi _1(M)$. This result strengthens analogous results in right-angled Artin groups and mapping class groups. Finally, we characterize hyperbolic strongly quasiconvex subgroups of a finitely generated 3-manifold group $\pi _1(M)$ by using their undistortedness property and their Morse elements.

Appendix A

1.1 Finite height subgroups, malnormal subgroups, and strongly quasiconvex subgroups of $\mathbb {Z}^k\rtimes _{\Phi } \mathbb {Z}$

In this section, we study strongly quasiconvex subgroups and finite height subgroups of abelian-by-cyclic subgroups $\mathbb {Z}^k\rtimes _{\Phi } \mathbb {Z}$. First, we define a fixed point of a group automorphism $\Phi :\!\mathbb {Z}^k \rightarrow \mathbb {Z}^k$ is a group element z in $\mathbb {Z}^k$ such that $\Phi (z)=z$. The main result of this section is the following proposition.

Proposition A.1

Let $\Phi :\!\mathbb {Z}^k \rightarrow \mathbb {Z}^k$ be a group automorphism. Then the group $G=\mathbb {Z}^k\rtimes _{\Phi } \mathbb {Z}=\langle \mathbb {Z}^k,t|tzt^{-1}=\Phi (z)\rangle$ has a finite height subgroup which is not trivial and has infinite index if and only if for every non-zero integer $\ell$, the group automorphism $\Phi ^{\ell }$ has no nontrivial fixed point. Moreover, a nontrivial, infinite index subgroup H has finite height if and only if H is a cyclic subgroup generated by a group element $g\in G-\mathbb {Z}^k$.

The proof of Proposition A.1 is a combination of Lemmas A.2, A.4, and A.5 as follows.

Lemma A.2

Let $G=\mathbb {Z}^k\rtimes _{\Phi } \mathbb {Z}=\langle \mathbb {Z}^k,t|tzt^{-1}=\Phi (z)\rangle$ and H a nontrivial subgroup of infinite index of G. Assume that H is a finite height subgroup. Then H is a cyclic subgroup generated by $t^{m}z$ where m is a positive integer and z is an element in $\mathbb {Z}^k$.

Proof

It follows from Corollary 2.5 that that $H\cap \mathbb {Z}^k$ is trivial. Thus H is a cyclic subgroup generated by $t^{m}z$ where m is a positive integer and z is an element in $\mathbb {Z}^k$. $\square$

Corollary A.3

(Strongly quasiconvex subgroups $\Longrightarrow$ are trivial or have finite index) Let $G=\mathbb {Z}^k\rtimes _{\Phi } \mathbb {Z}=\langle \mathbb {Z}^k,t|tzt^{-1}=\Phi (z)\rangle$ and H a strongly quasiconvex subgroup of G. Then either H is trivial or H has finite index in G.

Proof

We observe that the group G is a solvable group. By [11], none of the asymptotic cones of G has a global cut-point. Also by [10], the group G does not contain any Morse element.

Assume that H is not trivial and has infinite index in G. Then H is a finite height subgroup by Theorem 2.11. By Proposition A.2, H is a cyclic subgroup generated by $t^{m}z$ where m is a positive integer and z is an element in $\mathbb {Z}^k$. Therefore, G contains the Morse element $t^{m}z$ which is a contradiction. Thus, either H is trivial or H has finite index in G. $\square$

Lemma A.4

Let $\Phi :\!\mathbb {Z}^k \rightarrow \mathbb {Z}^k$ be a group automorphism such that $\Phi ^{\ell }$ has a nontrivial fixed point $z_0$ for some non-zero integer $\ell$. Let H be a finite height subgroup of $G=\mathbb {Z}^k\rtimes _{\Phi } \mathbb {Z}=\langle \mathbb {Z}^k,t|tzt^{-1}=\Phi (z)\rangle$. Then either H is trivial or H has finite index in G.

Proof

Assume that H is not trivial and has infinite index in G. By Proposition A.2, the subgroup H is a cyclic subgroup generated by $t^{m}z$ where m is a positive integer and z is an element in $\mathbb {Z}^k$. Since $\Phi ^{\ell }(z_0)=z_0$, the group element $z_0$ commutes to the group element $t^\ell$. Therefore, $z_0$ commutes to the group element $(t^{m}z)^\ell$ in H. Therefore, $\bigcap \nolimits _{i=1}^{\infty } z^{i}_0Hz^{-i}_0$ is infinite. Also $z^{i}_0H\ne z^{j}_0H$ for each $i\ne j$. Therefore, H is not a finite height subgroup of G which is a contradiction. Therefore, either H is trivial or H has finite index in G. $\square$

Lemma A.5

Let $\Phi :\!\mathbb {Z}^k \rightarrow \mathbb {Z}^k$ be a group automorphism such that $\Phi ^{\ell }$ has no nontrivial fixed point for every non-zero integer $\ell$. Let $G=\mathbb {Z}^k\rtimes _{\Phi } \mathbb {Z}=\langle \mathbb {Z}^k,t|tzt^{-1}=\Phi (z)\rangle$ and H be the cyclic subgroup of G generated by $t^{m}z$ where m is a positive integer and z is an element in $\mathbb {Z}^k$. Then H has height at most m. In particular, any cyclic group generated by tz where $z\in \mathbb {Z}^k$ is malnormal.

Proof

Assume that H does not have height at most m. Then there are $(m+1)$ distinct left cosets $g_1H, g_2H, g_3H,\ldots , g_{m+1}H$ such that $\bigcap \nolimits _{i=1}^{m+1}g_i H g^{-1}_i$ is infinite. We observer that there are $i\ne j$ such that $g=g^{-1}_ig_j$ can be written of the form $t^{mq}z_1$ for some integer q and some group element $z_1\in \mathbb {Z}$. Since $g=t^{mq}z_1$ is not a group element in H, we can write $g=z_0 (t^mz)^q$ for some group element $z_0\in \mathbb {Z}-\{e\}$.

Since the subgroup $g_iHg^{-1}_i\cap g_jHg^{-1}_j$ is infinite, the subgroup $H\cap gHg^{-1}$ is also infinite. Therefore, there is a non-zero integer p such that $g(t^mz)^pg^{-1}=(t^mz)^p$. Also, $g=z_0 (t^mz)^q$ for some group element $z_0\in \mathbb {Z}-\{e\}$. Thus, $z_0(t^mz)^pz_0^{-1}=(t^mz)^p$. It is straight forward that $(t^mz)^p=t^{mp}z'$ for some $z'\in \mathbb {Z}^k$. Therefore, $z_0(t^{mp}z')z_0^{-1}=t^{mp}z'$. This implies that $z_0(t^{mp})z_0^{-1}=t^{mp}$. Thus, $\Phi ^{mp}(z_0)=t^{mp}z_0t^{-mp}=z_0$ which is a contradiction. Therefore, H is a finite height subgroup. $\square$

Example A.6

Let $\Phi :\mathbb {Z}^2 \rightarrow \mathbb {Z}^2$ be an automorphism such that its corresponding matrix has the form $\Bigl (\begin{matrix} a&{}b \\ c&{}d \end{matrix} \Bigr )$ where $ad -bc =1$ and $|a+d| >2$. We note that $\phi$ has two real eigenvalues $\lambda$ and $1/\lambda$ such that $\lambda \ne 1,-1$ and $Trace(\Phi ) = a+ d = \lambda + 1/ \lambda$. For any non-zero integer $\ell$, the automorphism $\Phi ^{\ell }$ has two eigenvalues $\lambda ^{\ell }$ and $1/(\lambda )^{\ell }$ which have absolute value $\ne 1$. Hence $\Phi ^{\ell }$ has no nontrivial fixed point. Another way to see is that $\Phi ^{\ell }$ has the form of $\Bigl (\begin{matrix} a'&{}b' \\ c'&{}d' \end{matrix} \Bigr )$ where $a'd' -b'c' =1$. Since $Trace(\Phi ^{\ell }) = a'+d' = \lambda ^{\ell } + 1/(\lambda )^{\ell }$, it follows that $|a' +d'| >2$. It easy to see that the matrix $\Bigl (\begin{matrix} a'&{}b' \\ c'&{}d' \end{matrix} \Bigr )$ has no nontrivial fixed point (otherwise $|a'+d'|=2$). By Proposition A.1 and Corollary A.3, the group $Z^{2} \rtimes _\Phi \mathbb {Z}$ has a finite height subgroup H which is not strongly quasiconvex.

1.2 The $\mathcal {PS}$ system and strong quasiconvexity

In this section, we first give the proof for the statement that all special paths for a flip graph manifold are uniform quasi-geodesic. This fact seems not to be proved explicitly in [24, 25]. Then we give the proof for Lemma 3.6 which states that a $\mathcal {PS}$-contracting subset is strongly quasiconvex.

Proposition A.7

The special path for a flip graph manifold is uniform quasi-geodesic.

Let M be a flip manifold. Let $\tilde{M}_0 = \tilde{F}_{0} \times \mathbb {R}$ and $\tilde{M}_1 = \tilde{F}_{1} \times \mathbb {R}$ be two adjacent pieces in $\tilde{M}$ with a common boundary $\tilde{T}_0 = \tilde{M}_{0} \cap \tilde{M}_{1}$. Since M is a flip manifold, it follows that the boundary line $\overrightarrow{\ell _1}: = (\tilde{F}_{0} \times \{0\}) \cap \tilde{T}_{0}$ of $\tilde{F}_{0}$ projects to a fiber in $M_1$ and the boundary line $\overleftarrow{\ell _1}: = (\tilde{F}_{1} \times \{0\}) \cap \tilde{T}_{0}$ of $\tilde{F}_1$ projects to a fiber in $M_0$.

By abuse of language, we denote by $d_h$ the hyperbolic distance on ${\tilde{F}}_i$, and $d_v$ the fiber distance of ${\tilde{M}}_i$ for $i=0,1$. However, the following fact is crucial: the $d_h$-distance in ${\tilde{M}}_0$ on the boundary $\overrightarrow{\ell _1}$ of $\tilde{F}_0$ coincides with the $d_v$-distance on the fiber $\overleftarrow{\ell _1}$ of ${\tilde{M}}_1$.

Let $\delta =[x,y][y,z]$ be a concatenated path of geodesics [x, y] and [y, z] where $x=(x^h,x^v)\in {\tilde{M}}_0$, $y=(y^h,y^v)=(y^v, y^h)\in {\tilde{M}}_0\cap {\tilde{M}}_1={\tilde{T}}_0$, and $z=(z^h,z^v)\in {\tilde{M}}_1$. Note that the coordinates of y in ${\tilde{M}}_0$ and ${\tilde{M}}_1$ are switched.

Consider a minimizing horizontal slide of y in $\tilde{M}_0$ which changes its ${\tilde{F}}_0$-coordinate only so that the projection of [x, y] on ${\tilde{F}}_0$ is orthogonal to $\overrightarrow{\ell _1}$. To be precise, a minimizing horizontal slide in ${\tilde{M}}_0$ applied to y gives a point $w=(w^h,y^v)\in \tilde{T_0} \cap {\tilde{M}}_0$ with the same $\overleftarrow{\ell _1}$-coordinate as y so that $d_h(x^h, w^h)$ minimizes the distance $d_h(x^h, \overrightarrow{\ell _1})$.

We need the following observation that a minimizing horizontal slide does not increase distance too much. In what follows, we will work with $L^1$-metric on ${\tilde{M}}$ and denote by $|x-y|$ by the $L^1$-distance from x to y.

Lemma A.8

(Horizontal slide) There exists a constant $C>0$ depending only on M with the following property. If $w=(w^h,y^v)\in \tilde{T_1} \cap {\tilde{M}}_0$ is a point so that $d(x^h, w^h)$ minimizes the distance of $d(x^h, \overrightarrow{\ell _1})$, then

$$\begin{aligned}|x-w|+|w-z|\le |x-y| +|y-z|+ C\end{aligned}$$

Proof

We have

$$\begin{aligned}|x-w| + |w-z| = d_h(x^h, w^h)+d_h(y^v, z^h)+d_v(x^v, y^v)+d_v(w^h, z^v) \end{aligned}$$

and

$$\begin{aligned} |x-y| + |y-z| = d_h(x^h, y^h)+d_h(y^v, z^h)+d_v(x^v, y^v)+d_v(y^h, z^v). \end{aligned}$$

Hence, it suffices to find a constant C depending only on M such that the following inequality holds.

$$\begin{aligned} d_h(x^h, w^h) +d_v(w^h, z^v) \le d_h(x^h, y^h)+d_v(y^h, z^v)+C. \end{aligned}$$

(1)

By assumption, $w^h$ is a shortest projection point of $x^h$ to $\overrightarrow{\ell _1}$. By hyperbolicity of ${\tilde{F}}_0$, since $y^h$ and $w^h$ lie on $\overrightarrow{\ell _1}$, we have

$$\begin{aligned} d_h(x^h, w^h)+d_h(w^h, y^h)\le d_h(x^h, y^h) + C \end{aligned}$$

(2)

for some constant $C>0$ depending only on hyperbolicity constant of ${\tilde{F}}_0$. Since the $d_h$-distance in ${\tilde{M}}_0$ on the boundary $\overrightarrow{\ell _1}$ of ${\tilde{F}}_0$ coincides with the $d_v$-distance of ${\tilde{M}}_1$ on the fiber $\overrightarrow{\ell _1}$, we then obtain

$$\begin{aligned}d_h(w^h, y^h)=d_v(w^h, y^h)=|d_v(w^h, z^v)-d_v(y^h, z^v)|,\end{aligned}$$

which with (2) together proves (1). The lemma is thus proved. $\square$

Proof of the Proposition A.7

We use the notion $a \asymp b$ if there exists $K = K(M)$ such that $a/ K \le b \le Ka$.

We follow the notations in Definition 3.7. Let $\gamma =\gamma _0\cdot \gamma _1 \ldots \gamma _n$ be a special path between $x = x_0 \in {\tilde{M}}_0$ and $y= x_{n+1}\in {\tilde{M}}_{n}$ so that $\gamma _i=[x_i, x_{i+1}]\subset {\tilde{M}}_i$ where $x_{i+1}\in {\tilde{T}}_i:={\tilde{M}}_i\cap {\tilde{M}}_{i+1}$ for $0\le i \le n-1$. Then we have

$$\begin{aligned} \ell (\gamma ) = \sum _{i=0}^n d(x_i, x_{i+1}) \asymp \sum _{i=0}^n |x_i- x_{i+1}| \end{aligned}$$

where $\asymp$ follows from the fact that the $L^1$-metric is bi-lipschitz equivalent to a $L^2$-metric on each piece.

Let $\delta$ be the ${{\,\mathrm{CAT}\,}}(0)$ geodesic with same endpoints as $\gamma$. Let $y_i$ be the intersection point of $\delta$ with ${\tilde{T}}_{i-1}$. We let $y_0: = x_0$ and $y_{n+1} := x_{n+1}$. Then

$$\begin{aligned} \ell (\delta ) = \sum _{i=0}^n d(y_i, y_{i+1})\asymp \sum _{i=0}^n |y_i- y_{i+1}| \end{aligned}$$

We apply a sequence of minimizing horizontal slides to the endpoints of geodesics $[y_i, y_{i+1}]$ in order to transform the geodesic $\delta$ to the special path $\gamma$. More precisely, we first apply a horizontal slide (in $\tilde{M}_0$) of $y_1\in {\tilde{T}}_0$ to $w_1$ so by Lemma A.8, we have

$$\begin{aligned} |y_0 - w_1| + |w_1 - y_2| \le |y_0 - y_1| + |y_1 -y_2| + C \end{aligned}$$

and then apply a horizontal slide of $w_1$ to $x_1$ in $\tilde{M}_1$ we have

$$\begin{aligned} |y_2 - x_1| + |x_1 - y_0| \le |y_2 -w_1| + |w_1 -y_0| + C \end{aligned}$$

Using the fact $y_0 = x_0$ and two inequalities above, we have

$$\begin{aligned} |x_0 -x_1| = |y_0 - x_1| \le |y_0 -x_1|+ |x_1 -y_2| \le |y_0-y_1|+ |y_1 -y_2| +2C \end{aligned}$$

Inductively, we apply horizontal slides of $y_i$ to $w_i$ in the piece ${\tilde{M}}_{i-1}$ and then $w_i$ to $x_i$ in the piece $\tilde{M}_{i}$ to get

$$\begin{aligned} \sum _{i=0}^{n}|x_i -x_{i+1}| \le 2\sum _{i=0}^{n}|y_i -y_{i+1}| + 2C(n+1) \end{aligned}$$

Let $\rho >0$ be the minimal distance of any two JSJ planes. Since $x_0 \in \tilde{M}_0$ and $x_{n+1} \in \tilde{M}_{n+1}$, it follows that $(n-1)\rho \le d(x,y)$. Hence

$$\begin{aligned} \sum _{i=0}^{n}|x_i -x_{i+1}|&\le 2\sum _{i=0}^{n}|y_i -y_{i+1}| + 2C(n+1)\\&=2\sum _{i=0}^{n}|y_i -y_{i+1}| + 2C(n-1) + 4C\\&\le 2\sum _{i=0}^{n}|y_i -y_{i+1}| + 2C d(x,y)/\rho + 4C \end{aligned}$$

Therefore, we showed that there exists $\kappa >0$ such that for any special path $\gamma$ in $\tilde{M}$ we have $\ell (\gamma ) \le \kappa d(\gamma _{+}, \gamma _{-}) + \kappa$, where $\gamma _{+}$ and $\gamma _{-}$ are endpoints of $\gamma$. The proposition is proved. $\square$

Now we give a proof for Lemma 3.6. Before giving the proof, we need the following fact. Recall that all paths in the $\mathcal {PS}$ system are c-quasi-geodesic for some uniform constant c.

Lemma A.9

[25, Lemma 2.4(3,4)] Let $\pi$ be a $\mathcal {PS}$-projection with constant C on $A \subset X$. Then there exists a constant $k=k(c, C)>0$ with the following properties:

• 1. For each $x \in X$ we have $diam(\pi (B_r(x))) \le C$ for $r = d(x, A)/k - k$.

• 2. For each $x\in X$ we have $d(x,\pi (x))\le kd(x,A)+k.$

Proof of the Lemma 3.6

Let $\gamma$ be a $(\lambda , \lambda )$-quasi-geodesic with two endpoints in A, where $\lambda \ge 1$. Let $\overline{c}=\max \{k, \lambda \}$. For a constant $R>0$, we consider a connected component $\alpha$ of $\gamma {-} N_R(A)$ with initial and terminal endpoints $\alpha _-, \alpha _+$. We need at most ${\overline{c}} \ell (\alpha )/(R-{\overline{c}}^2)$ balls of radius $R/{\overline{c}}-{\overline{c}}$ to cover $\alpha$, where $\ell (\alpha )$ denotes the length of $\alpha$.

We now set $R={\overline{c}}^2(1+2C).$ On the one hand, we obtain by Lemma A.9(1) that

$$\begin{aligned}diam(\pi (\alpha ))\le {\overline{c}} C \ell (\alpha )/(R-{\overline{c}}^2)\le \ell (\alpha )/2{\overline{c}}.\end{aligned}$$

On the other hand, since $\alpha$ is a $(\lambda ,\lambda )$-quasi-geodesic for $\lambda \le {\overline{c}}$, we have

$$\begin{aligned} \ell (\alpha )&\le \lambda d(\alpha _-,\alpha _+)+\lambda \\&\le {\overline{c}}(d(\alpha _-, \pi (\alpha _-))+d(\alpha _+, \pi (\alpha _+))+diam(\pi (\alpha )))+{\overline{c}}\\&\le {\overline{c}}(2{\overline{c}}R+2{\overline{c}}+ \ell (\alpha )/2{\overline{c}})+{\overline{c}}, \end{aligned}$$

where the last two inequalities follow from a projection estimate combined with Lemma A.9(2). We thus obtain $\ell (\alpha )\le 4{\overline{c}}^2 (R+2)$. As a consequence, we have that $\gamma$ is contained in the $4{\overline{c}}^2(R+2)$-neighborhood of A. This proves the lemma. $\square$