TITLE: Lower bound for spectral radius on $\operatorname{GL}(n,\mathbb{Z})$
QUESTION [8 upvotes]: Consider the group of matrices $G =\operatorname{GL}(n,\mathbb{Z})$ with integer entries and determinant $\pm 1$. For each matrix $D \in G$, the product of the eigenvalues of $D$ is equal to $\det D =\pm 1$, and so the spectral radius $\rho(D)$, which is the size of the largest eigenvalue, is at least one. Moreover, if $\rho(D) =1$, then all the eigenvalue have size 1 and can be proven to be roots of unity.
My question is this: is there a lower bound $B>1$ such that if $\rho(D) \neq 1$ then $\rho(D) \geq B$? If so, does this $B$ depend on $n$?

REPLY [6 votes]: Lehmer's conjecture (which is what you'll want to Google) says that the spectral radius $R$ of a non-cyclotomic polynomial of degree $n$ satisfies
$$ R > 1 + \frac{c}{n} $$
for an absolute constant $c$. There is even a conjecture for the value of $c$ corresponding to a certain 10th degree polynomial. Dobrowolski proved that
$$ R > 1 + \frac{c'}{n}\left(\frac{\log\log n}{\log n}\right)^3, $$
and as Doug Lind noted, it is known one can take $c'=1/2$, but no one has improved the form of Dobrowolski's result in more then 30 years. For non-reciprocal polys, Chris Smyth proved Lehmer's conjecture with best possible constant in that case.

REPLY [5 votes]: Using Qiaochu's terminology, there is an explicit lower bound for the  spectral radius $R$ of a non-cyclotomic polynomial of degree $n$, namely 
$$
R>1+\frac{1}{2n}\Bigl(\frac{\log\log n}{\log n}\Bigr)^3
$$
(see A. Dubickas, On a conjecture of A. Schinzel and H. Zassenhaus, Acta Arith. 63 (1993), 15-20 for a slightly sharper result).