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Computing varieties of representations of hyperbolic 3-manifolds into \(\text{SL}(4,\mathbb R)\). (English) Zbl 1117.57016

A hyperbolic structure on a closed orientable \(3\)-manifold is given by a discrete faithful representation \(\phi_0\) of its fundamental group into \(\text{SO}^+(3,1)\). Mostow rigidity shows that such a representation is unique, up to conjugacy, but one may seek deformations of \(\phi_0\) by enlarging the target group. The paper under review concerns the problem of computing the deformations into \(\text{SL}(4,\mathbb{R})\). In other work [Flexing closed hyperbolic \(3\)-manifolds, preprint], the authors examine the relation of \(\text{SL}(4,\mathbb{R})\)-deformations to families of discrete faithful representations into \(\text{PU}(2,1)\), and to families of real projective structures on the manifolds.
In the work described in the paper under review, the authors find that for the first 4500 \(2\)-generator closed manifolds of the Hodgson-Weeks census, “\(61\) admit nontrivial infinitesimal deformations. Of these, \(21\) have been shown to admit actual deformations, and there is compelling numerical evidence that a further \(31\) do. Of the remaining \(9\) manifolds, \(3\) have been proven to be rigid using a certain third-order obstruction \(\ldots\) and numerical evidence strongly suggests that the remaining \(6\) are rigid.” More detailed results for these \(61\) manifolds are summarized in a table in the paper. Since these manifolds are too small in volume to contain imbedded totally geodesic surfaces, their deformations cannot arise by bending, and the underlying reason for their existence remains a mystery.
The computational task is to find an imbedded copy \(\mathcal{V}\) of the character variety in the full variety of \(\text{SL}(4,\mathbb{R})\)-representations. This \(\mathcal{V}\) is described by a single representation \(\Psi\) sending the generators to matrices whose entries are algebraic functions of \(n\) independent parameters \(u_1,\dots,u_n\). Individual representations to \(\text{SL}(4,\mathbb{R})\) can then be obtained by evaluating \(\Psi\) at specifc points \((u_1,\dots,u_n)\) of the parameter space.
The process to compute \(\Psi\) has quite a few steps, and uses an assortment of computational techniques, along with some existing software packages. Here is a rough summary, which fails to do justice to the many ideas ingeniously combined in the work. Start with \(\phi_0\) accurate to a few decimal places, given by Snappea. Improve the approximation to very high accuracy by means of Newtonian iteration, and use it to compute the dimension of the space of essential infinitesimal deformations. By varying the entries of the generating pair, seek a candidate for a nontrivial deformation. If one is found, examine the traces of generators as one moves away from \(\phi_0\), and discern parameters \(u_1,\dots,u_n\) for which the traces appear to be algebraic functions over the field \(\mathbb{Q}(u_1,\dots,u_n)\). The next steps use polynomial interpolation over varying points of the parameter space, taking advantage of the fact that certain expressions in the traces and the entries of the representation matrices must satisfy linear dependencies with integer coefficients. Powerful software packages are available for solving such dependencies. After finding algebraic generators, first for the trace field and then for the field generated by the matrix entries of a pair of generators, it is possible to work out expressions for the entries of \(\Psi\). The group relations can then be formally verified for \(\Psi\) to ensure that true deformations have been found. The details of this procedure are given for the manifold \(\text{Vol3}\), obtained by \((3,1)\)-surgery on the cusped manifold \(\text{m}007\).
The authors note that the manifold \(\text{v}2678(2,1)\) apparently has a \(5\)-dimensional space of essential infinitesimal deformations, with the \(\text{SL}(4,\mathbb{R})\)-character variety having two \(3\)-dimensional branches meeting in a \(1\)-dimensional subvariety containing the character of \(\phi_0\). This manifold is probably the only case among the first 2000 manifolds of the census which admits deformations into \(\text{SO}(4,1)\).

MSC:

57M50 General geometric structures on low-dimensional manifolds
57-04 Software, source code, etc. for problems pertaining to manifolds and cell complexes
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