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Does there exist closed hyperbolic three manifold which is locally rigid in the space of its all conformally flat structures? If so could someone provide examples?

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    $\begingroup$ Could you say what is meant by a "deformation" and by a "trivial deformation", and also clarify whether you want an example of a closed manifold? $\endgroup$ – Igor Belegradek Mar 28 at 22:17
  • $\begingroup$ A charitable reading of your question is "Are there examples of compact Haken hyperbolic 3-manifolds where the hyperbolic structure is locally rigid among all flat conformal structures." If this is indeed your question then such examples do exist and are due to Kevin Scannell. $\endgroup$ – Misha Mar 29 at 1:15
  • $\begingroup$ See Theorem 2 of this paper by @Misha link.springer.com/article/10.1007%2FBF01459788 $\endgroup$ – Ian Agol Mar 29 at 3:32
  • $\begingroup$ Thank you @Misha for making my question appropriate.I have edited now and used your sentence. $\endgroup$ – Gorapada Bera Mar 29 at 4:13
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  1. Let $M$ be a compact hyperbolic 3-manifold and let $C(M)$ be the moduli space of flat conformal structures on $M$. The hyperbolic metric on $M$ defines a distinguished point $h\in C(M)$. One says that $h$ is locally rigid (in $C(M)$) if $h$ is an isolated point of $C(M)$. By Thurston's holonomy theorem, local rigidity of $h$ is equivalent to local rigidity of the holonomy representation $\rho_h$ of $h$ in $$ Rep(M):= Hom(\pi_1(M), O(4,1))/O(4,1).$$

  2. Let $\Gamma< O(3,1)$ be the image of $\rho_h$ (it is a uniform torsion-free lattice in $O(3,1)$). The group $O(3,1)$ and, hence, $\Gamma$, acts naturally on the Lorentz space ${\mathbb R}^{3,1}$. A sufficient condition for local rigidity of $\rho_h$ in $Rep(M)$ is vanishing of the 1st cohomology group $$ H^1(M; {\mathcal F})\cong H^1(\Gamma; {\mathbb R}^{3,1}), $$ where ${\mathcal F}$ is a local system on $M$ determined by the action of $\Gamma$ on ${\mathbb R}^{3,1}$.

A representation $\rho_h$ is called infinitesimally rigid if this cohomology group vanishes.

Here is what is known:

Theorem. There are compact oriented hyperbolic 3-manifolds $M$ for which $H^1(M; {\mathcal F})=0$. Hence, for such manifolds, $h$ is locally rigid in $C(M)$.

The earliest examples (or, rather, existence of examples) were obtained as Dehn surgery on hyperbolic 2-bridge knots, [1]: Infinitely many Dehn surgeries yield infinitesimally rigid manifolds. These manifolds were non-Haken. Later, examples of infinitesimally rigid Haken hyperbolic manifolds (where incompressible surface is quasifuchsian) were constructed in [2]. In [5] it was proven that for every 2-bridge knot all but finitely many Dehn surgeries yield infinitesimally rigid manifolds. In [6], the authors made a comprehensive numerical study of $H^1(M; {\mathcal F})$ for hyperbolic 3-manifolds. They determined that out of the first 4500 2-generator manifolds in the Hodgson–Weeks census, only 61 are not infinitesimally rigid.

Here is what is unknown:

Question 1. Are there compact hyperbolic non-Haken 3-manifolds $M$ for which $h$ is not locally rigid in $C(M)$?

Question 2. Are there compact hyperbolic 3-manifolds $M$ for which $H^1(M; {\mathcal F})\ne 0$ but $h$ is locally rigid in $C(M)$?

Given a nonzero cohomology class $c\in H^1(M; {\mathcal F})$, there is an infinite sequence of obstructions for "integrating" $c$ into a curve in $Rep(M)$ tangent to $c$. The first of these obstructions (the cup product) is known to vanish, [2], but higher obstructions (Massey products) are a mystery. There is a compelling evidence (coming from the study of projective deformations of hyperbolic structures, [7]) that Question 2 has positive answer.

Questions 1 and 2 are closely related since Scannell, [3], proved that for a non-Haken "Fibonacci" manifold $M$, $H^1(M; {\mathcal F})\ne 0$. We still do not know local rigidity in this case.

Question 3. Is there a compact hyperbolic $3$-manifold $M$ for which $C(M)=\{h\}$, i.e. $h$ is globally rigid?

Most likely, Dehn surgeries on 2-bridge knots have this property but it's hard to prove.

Question 4. What topological/geometric features of a hyperbolic 3-manifold $M$ are responsible for vanishing/nonvanishing of $H^1(M; {\mathcal F})$, resp. local rigidity/flexibility of the hyperbolic structure?

It was realized early on (first, by Bill Thurston) that any closed embedded totally-geodesic hypersurface in $M$ yields nontrivial conformally flat deformations of the hyperbolic metric. On the other hand, as it was proven by Scannell, having merely an incompressible quasifuchsian surface is not enough. One can conjecture that if $M$ contains an embedded quasifuchsian surface which is "almost totally geodesic" (as in the work of Kahn and Marković), then $h$ is not locally rigid in $C(M)$. A weaker form of this conjecture is:

Conjecture. Let $M$ be a compact hyperbolic 3-manifold. Then $M$ admits a sequence of finite coverings $M_i\to M$ such that ranks of $H^1(M_i; {\mathcal F})$ diverge to infinity.

This is known to be true for some classes of arithmetic manifolds (originally, for the simplest type, due to Millson, but also for other arithmetic types by Clozel and others), but is unknown for arbitrary arithmetic hyperbolic 3-manifolds.

Question 5. What is the "right analogue" of Thurston's measured geodesic laminations which are responsible for nontrivial deformations of the hyperbolic structure $h$ in $C(M)$?

The evidence points towards some "unidentified laminar objects" which, as sets, are disjoint unions of complete geodesics and of totally-geodesic surfaces with geodesic boundary; they should be equipped with a kind of transverse metric-measure structure. See [8] for some partial results in this direction.

[1] M. Kapovich, Deformations of representations of discrete subgroups of SO(3,1), Math. Ann. 299 (1994), 341–354.

[2] M. Kapovich and J. J. Millson, On the deformation theory of representations of fundamental groups of compact hyperbolic 3-manifolds, Topology 35 (1996), 1085–1106.

[3] K. Scannell, Infinitesimal deformations of some SO(3,1) lattices. Pacific J. Math. 194 (2000), no. 2, 455–464.

[4] K. Scannell, Local rigidity of hyperbolic 3-manifolds after Dehn surgery. Duke Math. J. 114 (2002), no. 1, 1–14.

[5] S. Francaviglia, J. Porti, Rigidity of representations in SO(4,1) for Dehn fillings on 2-bridge knots. Pacific J. Math. 238 (2008), no. 2, 249–274.

[6] D. Cooper, D. Long, M. Thistlethwaite. Computing varieties of representations of hyperbolic 3-manifolds into SL(4,ℝ). Experiment. Math. 15 (2006), no. 3, 291–305.

[7] D. Cooper, D. Long, M. Thistlethwaite. Flexing closed hyperbolic manifolds. Geom. Topol. 11 (2007), 2413–2440.

[8] A. Bart, K. Scannell, A note on stamping. Geom. Dedicata 126 (2007), 283–291.

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  • $\begingroup$ Again thank you so much @Misha for a very beautiful exposition of what I was looking for. I am very grateful. $\endgroup$ – Gorapada Bera Apr 1 at 5:34

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