Flexing closed hyperbolic manifolds

Abstract

We show that for certain closed hyperbolic manifolds, one can nontrivially deform the real hyperbolic structure when it is considered as a real projective structure. It is also shown that in the presence of a mild smoothness hypothesis, the existence of such real projective deformations is equivalent to the question of whether one can nontrivially deform the canonical representation of the real hyperbolic structure when it is considered as a group of complex hyperbolic isometries. The set of closed hyperbolic manifolds for which one can do this seems mysterious. 57M50

Constructing non-congruence subgroups of flexible hyperbolic 3-manifold groups

“…In [3], the authors showed that certain closed hyperbolic 3-manifolds have the property that their hyperbolic structure can actually be non-trivially deformed when considered as a real projective structure. We described such manifolds as flexible.…”

“…This paper shows that for flexible hyperbolic manifolds, one can explicitly construct a cofinal family of non-congruence subgroups; moreover, since this does not rely on [6] in any way, it gives a new proof of the failure of the congruence subgroup property in the flexible case. Furthermore, non-arithmetic flexible manifolds are known [3], so our method applies in cases where Lubotzky's does not. We give a rather explicit construction which shows: Theorem 1.1.…”

“…A (very) special case of a flexible manifold is the situation of a hyperbolic 3-manifold which contains an immersed totally geodesic closed surface; it is shown in [3] that such a manifold has a finite cover which is flexible. It follows that: In this more classical form, it seems clear that the result was known to Weisfeiler (see his remarks in the first few paragraphs of §11 of [11]).…”

“…However, if one uses the Klein model for hyperbolic space, one can regard this hyperbolic structure as a real projective structure, and as such, it turns out that for certain closed hyperbolic 3-manifolds, one can make non-trivial deformations [3]. Such manifolds are called flexible.…”

“…When M is a closed orientable hyperbolic 3-manifold, the hyperbolic structure gives rise to a canonical discrete faithful representation of its fundamental group ρ : π 1 (M ) −→ SO (3,1), unique up to conjugacy by Mostow rigidity. Denoting the image group by Γ, one sees easily from rigidity and finite generation that in fact Γ is conjugate to a subgroup of SO (3, 1; R), where R is obtained from the ring of integers of a number field by inverting a certain finite number of elements.…”