The discrete subgroups and Jørgensen's inequality for \(\mathrm{SL}(m,\mathbb C_p)\)
The main goal of this paper is to study discrete subgroups of \(\mathrm{SL}(m,\mathbb Q_p)\) and \(\mathrm{SL}(m,\mathbb C_p)\) and to give versions of Jørgensen's inequality, a type of discreteness criterion, when \(m=2\). Matrices in \(\mathrm{SL}(2,\mathbb C_p)\) can be divided into three groups based on their eigenvalues: A single eigenvalue of multiplicity two (parabolic), distinct eigenvalues with the same absolute value (elliptic), and distinct eigenvalues with different absolute values (hyperbolic). A key idea of this paper is to similarly group matrices in \(\mathrm{SL}(m,\mathbb C_p)\): All eigenvalues have absolute value one and the matrix cannot be conjugated to a diagonal matrix (parabolic), all eigenvalues have absolute value one and the matrix can be conjugated to a diagonal matrix (elliptic), and there exists at least one eigenvalue with absolute value not equal to one (loxodromic). Some main results are the following. A subgroup of \(\mathrm{SL}(m,\mathbb C_p)\) is discrete if and only if it does not contain any parabolic elements or any elliptic elements of infinite order. Let \(\mathbb K_p\) be a finite extension of \(\mathbb Q_p\). If a discrete subgroup of \(\mathrm{SL}(2,\mathbb K_p)\) contains elliptic elements of finite order only, then it is finite. Interestingly, this result is not true if we replace \(\mathbb K_p\) with \(\overline{\mathbb Q}_p\), which can be seen by taking the subgroup generated by all diagonal matrices with entries \(\zeta_{p^i},\;\zeta_{p^i}^{-1}\) where \(\zeta_{p^i}\) is a primitive \(p^i\)-th root of unity. In the final section of this paper, the authors extend work of \textit{J. V. Armitage} and \textit{J. R. Parker} [Prog. Math. 265, 97--111 (2008; Zbl 1196.30034)] to give a sharper version of Jørgensen's inequality for \(\mathrm{SL}(2,\mathbb Q_p)\). For example, suppose that if \(A \neq -I\) and \(B\) are elements of \(\mathrm{SL}(2,\mathbb Q_p)\) such that \(B\) neither fixes nor interchanges the fixed points of \(A\) and that these matrices generate a discrete subgroup. Then if \(p>3\) then \(\min\{|\mathrm{tr}^2(A) - 4|, |\mathrm{tr}([A,B]) -2| \} \geq 1\), and when \(p\) is \(2\) or \(3\), the \(1\) can be replaced by 1/3 and 1/4, respectively. A version of Jørgensen's inequality for \(\mathrm{SL}(m,\mathbb C_p)\) is also given. The results are clearly stated and the proofs are not difficult to read.
scientific article; zbMATH DE number 6142985