In an informal talk I heard a statement:

"Any cyclic subgroup in a linear group is at most exponentially distorted"

with a vague reference to a work of Lubotzky with coauthors.

The works of Lubotzky that may be relevant to this quiestion and that I was able to find reference for:

Lubotzky, Alexander; Mozes, Shahar; Raghunathan, M. S. Cyclic subgroups of exponential growth and metrics on discrete groups. C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 8, 735–740. http://www.ams.org/mathscinet-getitem?mr=1244421

Lubotzky, Alexander; Mozes, Shahar; Raghunathan, M. S. The word and Riemannian metrics on lattices of semisimple groups. Inst. Hautes Études Sci. Publ. Math. No. 91 (2000), 5–53 (2001). http://www.ams.org/mathscinet-getitem?mr=1828742

seem to deal with lattices in Lie groups, not with arbitrary linear groups. So the question is:

Is the statement above true in such generality? If not, then for which classes of linear groups cyclic subgroups are at most exponentially distorted?

EDIT: As far as I understood, the distortion of a cyclic subgroup $H$ inside $G\subset GL(n,F)$ is meant here to be with respect to the respective word metrics on $H$ and $G$, not the underlying metric on $GL(n,F)$.

EDIT2: Sorry for the possible confusion. I guess, the distortion here is meant in the following sense:

Let $H$ be a subgroup of $G$ and fix some generating sets for $H$ and $G$. Consider a ball $B(n,1)$ of radius $n$ in the Cayley graph of $G$ centered at $1$, i.e. the set of all elements of $G$ having word length $\le n$ with respect to the generating set of $G$. Define a function $$ f(n):=\max\{|h|_H \mid h\in H\cap B(n,1)\}, $$ where $|h|_H$ is the length of the element $h$ with respect to the generating set of $H$.

Then $f(n)$ measures the distortion of subgroup $H$ inside $G$, and its order of growth is independent on the choice of the generating sets.

For example, in the group $G=\langle a,b\mid aba^{-1}=b^2\rangle$, subgroup $H=\langle b\rangle$ has exponential distortion function, as the element $h=a^mba^{-m}=b^{2^m}$ lies in the ball $B(2m+1,1)$ in $G$, but the word length of $g$ in $H$ is $2^m$.

But if we consider a group $G=\langle a,b,c\mid aba^{-1}=b^2, bcb^{-1}=c^2\rangle$ then a similar reasoning shows that $H=\langle c \rangle$ has double exponential distortion in $G$: $f(n)=2^{2^n}$.

The statement I am inquiring about stipulates that cyclic subgroups $H$ of linear groups $G\subset GL(n,F)$ cannot have distortion functions that grow faster than exponential functions.