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> Let $a,b$ be two loxodromic isometries of a CAT(0) space. Assume that, for every $n \geq 1$, $a^nb$ is also loxodromic. Is it possible for the translation length of $a^nb$ to be bounded independently of $n$?
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First, I thought as obvious that the translation length of $a^nb$ has to tend to $+ \infty$ as $n \to + \infty$, but I may have been misled by the CAT(-1) case (where this is clearly true). Now, I go back and forth between a possible counterexample and an easy argument I am missing...



Translation lengths in CAT(0) spaces
Consider the following two transformations of $\mathbb{R}^3$: $a:(x,y,z)\mapsto (x+1,y,z)$ and $b:(x,y,z)\mapsto (-x,-y,z+1)$. The translation axis for $a^nb$ is the line $x=n/2$, $y=0$ and $a^nb$ translates this line by a distance of 1, independent of $n$.


