Abstract
In this paper, we study strongly quasiconvex subgroups in a finitely generated 3manifold group \(\pi _1(M)\). We prove that if M is a compact, orientable 3manifold that does not have a summand supporting the Sol geometry in its spheredisc 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 spheredisc 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 BassSerre tree of \(\pi _1(M)\). This result strengthens analogous results in rightangled Artin groups and mapping class groups. Finally, we characterize hyperbolic strongly quasiconvex subgroups of a finitely generated 3manifold group \(\pi _1(M)\) by using their undistortedness property and their Morse elements.
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References
 1.
Agol, I.: Tameness of hyperbolic 3manifolds. Mathematics (2004). arXiv:math/0405568
 2.
Bestvina, M., Bromberg, K., Kent, A.E., Leininger, C.J.: Undistorted purely pseudoanosov groups (2016). arXiv:1608.01583
 3.
Bestvina, M.: Questions in geometric group theory. M. Bestvina’s home page (2004)
 4.
Bridson, M.R., Haefliger, A.: Metric spaces of nonpositive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1999)
 5.
Behrstock, J.A., Neumann, W.D.: Quasiisometric classification of graph manifold groups. Duke Math. J. 141(2), 217–240 (2008)
 6.
Bigdely, H., Wise, D.T.: Quasiconvexity and relatively hyperbolic groups that split. Mich. Math. J. 62(2), 387–406 (2013)
 7.
Canary, R.D.: A covering theorem for hyperbolic 3manifolds and its applications. Topology 35(3), 751–778 (1996)
 8.
Calegari, D., Gabai, D.: Shrinkwrapping and the taming of hyperbolic 3manifolds. J. Am. Math. Soc. 19(2), 385–446 (2006)
 9.
Dahmani, F.: Combination of convergence groups. Geom. Topol. 7, 933–963 (2003)
 10.
Druţu, C., Mozes, S., Sapir, M.: Divergence in lattices in semisimple Lie groups and graphs of groups. Trans. Am. Math. Soc. 362(5), 2451–2505 (2010)
 11.
Druţu, C., Sapir, M.: Treegraded spaces and asymptotic cones of groups. Topology 44(5), 959–1058 (2005) (With an appendix by Denis Osin and Sapir)
 12.
Durham, M.G., Taylor, S.J.: Convex cocompactness and stability in mapping class groups. Algebraic Geom. Topol. 15(5), 2839–2859 (2015)
 13.
Genevois, A.: Hyperbolicities in CAT(0) cube complexes. arXiv:1709.08843
 14.
Gersten, S.M.: Quadratic divergence of geodesics in \({\rm CAT}(0)\) spaces. Geom. Funct. Anal. 4(1), 37–51 (1994)
 15.
Gitik, R., Mitra, M., Rips, E., Sageev, M.: Widths of subgroups. Trans. Am. Math. Soc. 350(1), 321–329 (1998)
 16.
Kim, H.: Stable subgroups and Morse subgroups in mapping class groups (2017). arXiv:1710.11617
 17.
Kim, S.H., Koberda, T.: The geometry of the curve graph of a rightangled Artin group. Int. J. Algebra Comput. 24(2), 121–169 (2014)
 18.
Kapovich, M., Leeb, B.: 3manifold groups and nonpositive curvature. Geom. Funct. Anal. 8(5), 841–852 (1998)
 19.
Koberda, T., Mangahas, J., Taylor, S.J.: The geometry of purely loxodromic subgroups of rightangled Artin groups. Trans. Am. Math. Soc. 369(11), 8179–8208 (2017)
 20.
Leeb, B.: 3manifolds with(out) metrics of nonpositive curvature. Invent. Math. 122(2), 277–289 (1995)
 21.
Nguyen, H.T., Sun, H.: Subgroup distortion of 3manifold groups (2019). arXiv:1904.12253 (To appear in Trans. Amer. Math. Soc)
 22.
Osin, D.V.: Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems. Mem. Am. Math. Soc. 179(843), 1–100 (2006)
 23.
Przytycki, P., Wise, D.T.: Separability of embedded surfaces in 3manifolds. Compos. Math. 150(9), 1623–1630 (2014)
 24.
Sisto, A.: 3manifold groups have unique asymptotic cones (2011). arXiv:1109.4674
 25.
Sisto, A.: Contracting elements and random walks. J. Reine Angew. Math. 742, 79–114 (2018)
 26.
Sun, H.: A characterization on separable subgroups of 3manifold groups (2018). arXiv:1805.08580 (To appear in Journal of Topology)
 27.
Tran, H.C.: Malnormality and joinfree subgroups in rightangled Coxeter groups (2017). arXiv:1703.09032
 28.
Tran, H.C.: On strongly quasiconvex subgroups. Geom. Topol. 23(3), 1173–1235 (2019)
Acknowledgements
The authors are grateful for the insightful and detailed critiques of the referee that have helped improve the exposition of this paper. The authors especially appreciate the referee for pointing out a mistake in Theorem 1.4 in the earlier version. H. T. was supported by an AMSSimons Travel Grant. W. Y. is supported by the National Natural Science Foundation of China (No. 11771022).
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Appendix A
Appendix A
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 abelianbycyclic 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 \(G=\mathbb {Z}^k\rtimes _{\Phi } \mathbb {Z}=\langle \mathbb {Z}^k,ttzt^{1}=\Phi (z)\rangle \) has a finite height subgroup which is not trivial and has infinite index if and only if for every nonzero 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 \(G=\mathbb {Z}^k\rtimes _{\Phi } \mathbb {Z}=\langle \mathbb {Z}^k,ttzt^{1}=\Phi (z)\rangle \) 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 \(G=\mathbb {Z}^k\rtimes _{\Phi } \mathbb {Z}=\langle \mathbb {Z}^k,ttzt^{1}=\Phi (z)\rangle \) 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 cutpoint. 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 nonzero integer \(\ell \). Let H be a finite height subgroup of \(G=\mathbb {Z}^k\rtimes _{\Phi } \mathbb {Z}=\langle \mathbb {Z}^k,ttzt^{1}=\Phi (z)\rangle \). 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 nonzero integer \(\ell \). Let \(G=\mathbb {Z}^k\rtimes _{\Phi } \mathbb {Z}=\langle \mathbb {Z}^k,ttzt^{1}=\Phi (z)\rangle \) 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 nonzero 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 nonzero 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.
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 quasigeodesic. 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 quasigeodesic.
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 \(xy\) 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
Proof
We have
and
Hence, it suffices to find a constant C depending only on M such that the following inequality holds.
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
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
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 \(0\le i \le n1\). Then we have
where \(\asymp \) follows from the fact that the \(L^1\)metric is bilipschitz 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 \({\tilde{T}}_{i1}\). We let \(y_0: = x_0\) and \(y_{n+1} := x_{n+1}\). Then
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
and then apply a horizontal slide of \(w_1\) to \(x_1\) in \(\tilde{M}_1\) we have
Using the fact \(y_0 = x_0\) and two inequalities above, we have
Inductively, we apply horizontal slides of \(y_i\) to \(w_i\) in the piece \({\tilde{M}}_{i1}\) and then \(w_i\) to \(x_i\) in the piece \(\tilde{M}_{i}\) to get
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 \((n1)\rho \le d(x,y)\). Hence
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 cquasigeodesic 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 \((\lambda , \lambda )\)quasigeodesic 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
On the other hand, since \(\alpha \) is a \((\lambda ,\lambda )\)quasigeodesic for \(\lambda \le {\overline{c}}\), we have
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. \(\square \)
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Nguyen, H.T., Tran, H.C. & Yang, W. Quasiconvexity in 3manifold groups. Math. Ann. (2020). https://doi.org/10.1007/s0020802002044y
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