# Translation lengths in CAT(0) spaces

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 $$n$$?

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...

• Could you remind us of the definition of loxodromic?
– IJL
Mar 5, 2021 at 10:08
• Loxodromic = there exists a bi-infinite geodesic on which the isometry acts as a translation. Mar 5, 2021 at 10:41
• @PeterKosenko: You should write $\ell(a^nb)= \lim\limits_{k \to + \infty} d(x_0,(a^nb)^kx_0)/k$. Mar 5, 2021 at 11:13
• Yes, you are correct -- and now I realize that it doesn't immediately imply your statement... Mar 5, 2021 at 11:51

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 $$n$$.
• I've just realized that the $y$-coordinate isn't needed: $a$ and $b$ both preserve the plane $y=0$, so there is a simpler example in $\mathbb{R}^2$.
• This even works restricted to the $\mathbb{R}^2$ that is the xz-plane. Mar 5, 2021 at 16:59