Lehmer’s conjecture is one of the most striking open problems in number theory. It roughly postulates that the complex roots \(a\) with \(|a| > 1\) of a polynomial with integral coefficients cannot simultaneously be close to the unit circle (unless all non-zero roots lie on the unit circle). More precisely, the Mahler measure of a polynomial \(P \in \mathbb{Z}[t]\) which factors as \(P = c (t-a_1)(t-a_2)\cdots(t-a_n)\) is defined to be
\[M(P) = |c| \prod_{i=1}^n \max(|a_i|,1).\]
Lehmer’s conjecture states that there is a constant \(L > 1\) such that every integral polynomial satisfies \(M(P) = 1 \) or \(M(P) \geq L\). The conjectured value of \(L\) is \(1.17628…\) which is the Mahler measure of the polynomial \(t^{10} + t^9 − t^7 − t^6 − t^5 − t^4 − t^3 + t + 1\). The conjecture extends to Laurent polynomials, i.e. elements of \(\mathbb{Z}[t,t^{-1}]\).
The Laurent polynomial ring \(\mathbb{Z}[t,t^{-1}]\) can be interpreted as the group ring of the infinite cyclic group \(\mathbb{Z}\). In this way every Laurent polynomial gives rise to a multiplication operator on \(\ell^2(\mathbb{Z})\). The Mahler measure of a Laurent polynomial \(P \in \mathbb{Z}[t,t^{-1}]\) can be computed as
\[M(P) = \exp \int_{S^1} \ln(|p(z)|) dz\]
which is exactly the Fuglede-Kadison determinant of the associated multplication operator \(r_P \colon \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})\). This observation is the starting point of a paper of Lück which appeared on the arXiv in 2019 (https://arxiv.org/abs/1901.00827). In a nutshell Lück posed the intruiging question, whether Lehmer’s conjecuture might hold in greater generality:
Given a torsion-free group \(G\). Is the the Fuglede-Kadison determinant of elements in \(\mathbb{Z}[G]\) uniformly bounded away from \(1\) provided the value does not equal \(1\).
The restriction to torsion-free groups is due to the observation that large finite subgroups give rise to elements with very small Fuglede-Kadison determinant (see section 9 in Lück’s paper). In fact, Lück proposed additional variations of Lehmer’s conjecture for torsion-free groups. One should also mention that this problem is linked to the determinant conjecture: Is the Fuglede-Kadison determinant of elements in the integral group ring bounded below by 1? The determinant conjecture has been confirmed for all sofic groups.
A major problem concerning the generalized Lehmer problem is that Fuglede-Kadison determinants are hard to compute, so it is hard to find counterexamples or collect evidence. A geometric approach was suggested by Lück. The Fuglede-Kadison determinant is used to define the \(L^2\)-torsion of manifolds and the \(L^2\)-torsion of hyperbolic 3-manifolds can computed in terms of the volume. In particular, the fundamental groups of small volume hyperbolic 3-manifolds admit matrices over the integral group ring with small Fuglede-Kadison determinant. However, it is not clear if small Fuglede-Kadison determinants can also be achieved by a single element in the integral group ring.
Last week Fathi Ben Aribi posted a paper on the arxiv where he was able to make some progress. On the one hand he computed the Fuglede-Kadison determinant for simple elements in the integral group ring of non-abelian free groups. For instance, for \(F = \langle x,y \rangle\), he showed that \(\det(1 + x +y) = \frac{2}{\sqrt{3}} \approx 1,154…\). This value is already smaller than the expected value in the classical Lehmer conjecture. In additon, he was able to improve Lück’s geometric approach. Under suitable assumptions a small Fuglede-Kadison determinant for matrices over the group ring of \(\pi_1\) of a hyperbolic 3-manifold, gives rise to small Fuglede-Kadison determinants of elements in the integral group ring. This applies, in particular, to the Weeks manifold and implies that the Fuglede-Kadison determinant of a group ring element can take a value of \(\approx 1.051…\).
One may at least conclude that for general torsion-free groups the answer to the Lehmer problem is really different from the expected answer for polynomials.