Graph Manifold Groups Research Articles

Quasiconvexity in 3-manifold groups

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.

Characteristic varieties of graph manifolds and quasi-projectivity of fundamental groups of algebraic links

The present paper studies the structure of characteristic varieties of fundamental groups of graph manifolds. As a consequence, a simple proof solving a question posed by Papadima on the characterization of algebraic links that have quasi-projective fundamental groups is provided. The type of quasi-projective obstructions used here are in the spirit of Papadima’s original work.

On the coarse geometry of certain right-angled Coxeter groups

Let $\Gamma$ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph $\Gamma$ is $\mathcal{CFS}$, we prove that the right-angled Coxeter group $G_\Gamma$ is virtually a Seifert manifold group or virtually a graph manifold group and we give a complete quasi-isometry classification of these such groups. Otherwise, we prove that $G_\Gamma$ is hyperbolic relative to a collection of $\mathcal{CFS}$ right-angled Coxeter subgroups of $G_\Gamma$. Consequently, the divergence of $G_\Gamma$ is linear, or quadratic, or exponential. We also generalize right-angled Coxeter groups which are virtually graph manifold groups to certain high dimensional right-angled Coxeter groups (our families exist in every dimension) and study the coarse geometry of this collection. We prove that strongly quasiconvex torsion free infinite index subgroups in certain graph of groups are free and we apply this result to our right-angled Coxeter groups.

Profinite rigidity of graph manifolds and JSJ decompositions of 3-manifolds

There has been much recent interest into those properties of a 3-manifold determined by the profinite completion of its fundamental group. In this paper we give readily computable criteria specifying precisely when two orientable graph manifold groups have isomorphic profinite completions. Our results also distinguish graph manifolds among the class of all 3-manifolds and give information about the structure of totally hyperbolic manifolds, and give control over the pro-p completion of certain graph manifold groups.

Contracting elements and random walks

Abstract We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper CAT ⁢ ( 0 ) {\mathrm{CAT}(0)} spaces, elements acting hyperbolically on the Bass–Serre tree in graph manifold groups. We also define a related notion of weakly contracting element, and show that those coincide with hyperbolic elements in groups acting acylindrically on hyperbolic spaces and with iwips in Out ⁢ ( F n ) {\mathrm{Out}(F_{n})} , n ≥ 3 {n\geq 3} . We show that each weakly contracting element is contained in a hyperbolically embedded elementary subgroup, which allows us to answer a problem in [16]. We prove that any simple random walk in a non-elementary finitely generated subgroup containing a (weakly) contracting element ends up in a non-(weakly-)contracting element with exponentially decaying probability.

Residual properties of graph manifold groups

Let f : M → N be a continuous map between closed irreducible graph manifolds with infinite fundamental group. Perron and Shalen (1999) [16] showed that if f induces a homology equivalence on all finite covers, then f is in fact homotopic to a homeomorphism. Their proof used the statement that every graph manifold is finitely covered by a 3-manifold whose fundamental group is residually p for every prime p. We will show that this statement regarding graph manifold groups is not true in general, but we will show how to modify the argument of Perron and Shalen to recover their main result. As a by-product we will determine all semidirect products Z ⋉ Z d which are residually p for every prime p.

Quasi-isometric classification of graph manifold groups

We show that the fundamental groups of any two closed irreducible non-geometric graph-manifolds are quasi-isometric. This answers a question of Kapovich and Leeb. We also classify the quasi-isometry types of fundamental groups of graph-manifolds with boundary in terms of certain finite two-colored graphs. A corollary is the quasi-isometry classification of Artin groups whose presentation graphs are trees. In particular any two right-angled Artin groups whose presentation graphs are trees of diameter at least 3 are quasi-isometric, answering a question of Bestvina; further, this quasi-isometry class does not include any other right-angled Artin groups.

Separability criterion for graph-manifold groups

Some recent work in the theory of 3-manifolds and immersed surfaces indicates that the class of graph manifolds contains a large number of compact 3-manifolds whose fundamental groups are not subgroup separable (LERF). Rubinstein and Wang have given a criterion to determine whether or not a given horizontal surface in a graph manifold is separable (i.e., lifts to an embedded surface in some finite cover of the manifold). This paper extends the criterion to include some nonhorizontal surfaces in graph manifolds.