Non-LERFness of arithmetic hyperbolic manifold groups and mixed 3-manifold groups

Constructing Non-congruence Subgroups of Flexible Hyperbolic 3-manifold Groups

We give an explicit construction for non-congruence subgroups in the fundamental group of a flexible hyperbolic 3-manifold.

Isospectrality and 3-manifold Groups

In this note, we explain how a well-known construction of isospectral manifolds leads to an obstruction to a group being the fundamental group of a closed 3-dimensional manifold. The problem of determining, for a given group G, whether there is a closed 3-manifold M with π1(M) ∼= G is readily seen to be undecidable; let us write G ∈ G 3 if there is such a 3-manifold. A standard conjecture (rela...

Non-manifold Hyperbolic Groups of High Cohomological Dimension

Which 3-manifold Groups Are Kähler Groups?

The question in the title, first raised by Goldman and Donaldson, was partially answered by Reznikov. We give a complete answer, as follows: if G is both a (closed) 3-manifold group and a Kähler group, then G must be finite.

Kähler groups, quasi-projective groups and 3-manifold groups

We prove two results relating 3-manifold groups to fundamental groups occurring in complex geometry. Let N be a compact, connected, orientable 3-manifold. If N has non-empty, toroidal boundary, and π1(N) is a Kähler group, then N is the product of a torus with an interval. On the other hand, if N has either empty or toroidal boundary, and π1(N) is a quasi-projective group, then all the prime co...