Surface group amalgams that (don't) act on 3-manifolds

Abstract

We determine which amalgamated products of surface groups identified over multiples of simple closed curves are not fundamental groups of 3-manifolds. We prove each surface amalgam considered is virtually the fundamental group of a 3-manifold. We prove that each such surface group amalgam is abstractly commensurable to a right-angled Coxeter group from a related family. In an appendix, we determine the quasi-isometry classes among these surface amalgams and their related right-angled Coxeter groups.

Properly discontinuous actions versus uniform embeddings

“…Let G mn = π 1 (X mn ). Hruska, Stark, and Tran show the following: Theorem 6.4 (Theorem 5.6, [11]). For all m and n, G mn is virtually a 3manifold group.…”

“…We recall the examples of virtually 3-manifold groups constructed in [11] (the examples in [13] have similar proofs, which we explain in the next subsection). We start with two closed surfaces S a and S b of genus ≥ 2, and a choice of essential simple closed curves γ a and γ b on S a and S b respectively.…”

“…There are also examples where updim(G) is strictly less than actdim(G), but equal to actdim(H) for H a finite index subgroup of G. These examples are relatively easy to construct when G is allowed to have torsion, for example there are many virtually free groups which do not act properly on the plane (such as the free product of alternating groups A 5 * A 5 ). Torsion-free examples were constructed by Kapovich-Kleiner in [13] and Hruska-Stark-Tran in [11]. In both cases, the groups constructed were virtually 3-manifold groups but not 3-manifold groups.…”

“…Theorem 1.2. Let G be the k-fold direct product of the examples in [13] or [11], where the degree of the covering map is a multiple of 4. Then…”

Nonplanar graphs in boundaries of CAT(0) groups

“…Haïssinsky proved that the answer to Question 1.3 is positive if G is hyperbolic and cubulated [Haï15]. It is also necessary to ask for a finite-index subgroup, as there are torsion-free hyperbolic and CAT(0) groups with planar boundary which are not 3-manifold groups but have 3-manifold groups as finite-index subgroups [KK00,HST].…”