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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).

##### MSC:
 11G50 Heights 11R04 Algebraic numbers; rings of algebraic integers 22E40 Discrete subgroups of Lie groups
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