Arithmetic groups and Salem numbers
The author shows the equivalence of the following two open questions, one a natural question about arithmetic subgroups of \(SL(2,\mathbb{R})\), and the other a special case of a much studied problem of Lehmer: (1) Does there exist a neighbourhood \(W\) of the identity in \(SL(2,\mathbb{R})\) such that for every cocompact torsion-free arithmetic subgroup \(\Gamma\), we have \(\Gamma\cap W=\{e\}\)? (2) Does there exist a number \(\varepsilon>0\) such that any Salem number \(\tau\) satisfies \(\tau>1+\varepsilon\)? A Salem number is a real algebraic integer \(\tau>1\), all of whose conjugates lie inside the unit circle, with at least one conjugate on the unit circle. Alternatively, a Salem number is a number \(\tau>1\) for which \(\tau+\tau^{-1}\) is a totally real algebraic integer all of whose other conjugates lie in the interval \((-2,2)\). The smallest known Salem number is one of degree 10 which was discovered by Lehmer in 1933 and satisfies \(\tau=1.17628\dots\). The connection with cocompact arithmetic subgroups of \(SL(2,\mathbb{R})\) arises from the characterization of such subgroup in terms of orders in quaternion division algebras.
arithmetic subgroups of \(SL(2,\mathbb{R})\)
scientific article; zbMATH DE number 91452