  Hrube\v{s} and Wigderson [HW14] initiated the study of noncommutative
arithmetic circuits with division computing a noncommutative rational function
in the free skew field, and raised the question of rational identity testing.
It is now known that the problem can be solved in deterministic polynomial time
in the white-box model for noncommutative formulas with inverses, and in
randomized polynomial time in the black-box model [GGOW16, IQS18, DM18], where
the running time is polynomial in the size of the formula. The complexity of
identity testing of noncommutative rational functions remains open in general
(when the formula size is not polynomially bounded). We solve the problem for a
natural special case. We consider polynomial expressions in the free group
algebra $\mathbb{F}\langle X, X^{-1}\rangle$ where $X=\{x_1, x_2, \ldots,
x_n\}$, a subclass of rational expressions of inversion height one. Our main
results are the following. 1. Given a degree $d$ expression $f$ in
$\mathbb{F}\langle X, X^{-1}\rangle$ as a black-box, we obtain a randomized
$\text{poly}(n,d)$ algorithm to check whether $f$ is an identically zero
expression or not. We obtain this by generalizing the Amitsur-Levitzki theorem
[AL50] to $\mathbb{F}\langle X, X^{-1}\rangle$. This also yields a
deterministic identity testing algorithm (and even an expression reconstruction
algorithm) that is polynomial time in the sparsity of the input expression. 2.
Given an expression $f$ in $\mathbb{F}\langle X, X^{-1}\rangle$ of degree at
most $D$, and sparsity $s$, as black-box, we can check whether $f$ is
identically zero or not in randomized $\text{poly}(n,\log s, \log D)$ time.
