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A height gap theorem for finite subsets of \(\mathrm{GL}_{d}(\overline{\mathbb{Q}})\) and nonamenable subgroups. (English) Zbl 1243.11071
The author introduces a conjugation invariant height \(\hat{h}(F)\) on finite subsets of matrices \(F\) in \(\mathrm{GL}_{d}(\overline{\mathbb Q})\), where \(\overline{\mathbb{Q}}\) is the field of algebraic numbers, which is well suited to the study of geometric and arithmetic behavior of power sets \(F^{n} = F\cdot\cdot\cdot F\) for \(n\in\bar{\mathbb Q}\). Among many interesting results of the paper the following two theorems are outstanding: Let \(F\) be a finite subset of \(\mathrm{GL}_{d}(\overline{\mathbb Q})\) generating a nonamenable subgroup that acts strongly irreducibly. Then there exists a constant \(\varepsilon(d)>0\) such that \(\hat{h}(F)> \varepsilon(d)\). If \(F\) is a finite subset of \(\mathrm{GL}_{d}(\overline{\mathbb Q})\) generating a nonvirtually solvable subgroup, then there exists a constant \(\varepsilon(d)>0\) such that \(\hat{h}(F)> \varepsilon(d)\). Using these two theorems the author can for instance prove the following statements:
There is an \(n(d)\in\mathbb N\) such that if \(K\) is a field and \(F\) is a finite symmetric subset of \(\mathrm{GL}_{d}(K)\) containing \(1\) which generates a nonvirtually solvable subgroup, then \(F^n\) contains two elements \(A\) and \(B\) which generate a non-abelian free subgroup.
There is an an integer \(n(d)\in\mathbb N\) such that if \(K\) is a field and \(F\) is finite subset of \(\mathrm{GL}_{d}(K)\) which generates an infinite subgroup, then \((F\cup F^{-1})^n\) contains an element of infinite order.
The interpretation of \(\hat{h}(F)\) in terms of spectral radius allows the author to derive the following two statements:
There are constants \(n(d)\in\mathbb N\) and \(c(d)\in\mathbb N\) such that if \(F\) is any finite subset of \(\mathrm{GL}_{d}((\overline{\mathbb Q})\) containing \(1\), then there is some \(A\in F^n\) and some eigenvalue \(\lambda\) of \(A\) such that \[ h(\lambda)\geq \frac{1}{|F^{c}|}\cdot \hat{h}(F), \] where \(h(\lambda)\) is the height of \(\lambda\) in sense of [E. Bompieri and W. Gubler, Heights in Diophantic Geometry. New Mathematical Monographs 4. Cambridge: Cambridge University Press. (2006; Zbl 1115.11034)].
There are constants \(n(d)\in\mathbb N\) and \(\varepsilon(d)> 0\) such that if \(F\) is any finite subset of \(\mathrm{GL}_{d}(\overline{\mathbb Q})\) containing \(1\) and generating a nonvirtually solvable subgroup, then one finds a matrix \(A\in F^n\) and an eigenvalue \(\lambda\) of \(A\) such that \(h(\lambda)> \varepsilon(d)\).
If \(F\) is is a finite subset of \(\mathrm{GL}_{d}(\overline{\mathbb Q})\), then \(\hat{h}(F) = 0\) if and only if the group generated by \(F\) is virtually nilpotent.
The author introduces for \(F\) also the minimal height and compares the minimal height \(h(F)\) with the normalized height \(\hat{h}(F)\). Using estimates relating the minimal norm of \(F\) defined by the author and the matrix coefficients of the elements of \(F\) in the adjoint representation he is able to obtain a global upper bound on the height of the matrix coefficients of the finite set \(F\) of matrices in \(\mathrm{GL}_{d}(\overline{\mathbb Q})\). The author gives in the paper many hints at the relations of his results to classical statements (e.g., Margulis lemma, uniform Tits alternative, uniform version of the Burnside-Schur theorem) and to classical conjectures (e. g., the Lehmer problem).

11G50 Heights
11R04 Algebraic numbers; rings of algebraic integers
22E40 Discrete subgroups of Lie groups
Full Text: DOI
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