# Two definitions of horofunction for Gromov hyperbolic spaces

Let $$X$$ be a proper, geodesic, $$\delta$$-hyperbolic metric space (e.g. a hyperbolic group), and let $$x_0$$ be a basepoint for $$X$$. There seem to be two different definitions of "horofunction" for $$X$$, and I'd like to understand the relationship between them.

### First Definition

Definition 1. For each $$p\in X$$ let $$f_p\colon X\to\mathbb{R}$$ be the function $$f_p(x) = d(x,p)-d(x_0,p).$$ A function $$f\colon X\to \mathbb{R}$$ is called a horofunction if there exists an unbounded sequence $$\{p_n\}$$ in $$X$$ such that $$f_{p_n}$$ converges uniformly to $$f$$ on compact sets.

This definition is due to Gromov, and the set of all horofunctions on $$X$$ is known as the horofunction boundary. Note that this definition works for any metric space.

### Second Definition

The following definition seems to come out of the work of Coornaert and Papadopoulos on the symbolic dynamics of the visual boundary of a hyperbolic group, though it is similar to the "local" description of horofunctions using cocycles given by Gromov in his essay on hyperbolic groups.

Definition 2. A function $$f\colon X\to \mathbb{R}$$ with $$f(x_0)=0$$ is called a horofunction if it satisfies the following conditions:

1. There exists an $$\epsilon>0$$ so that $$f$$ is $$\epsilon$$-convex, in the sense that $$f(\gamma_t)\leq (1-t)f(\gamma_0) + t f(\gamma_1) + \epsilon$$ for every constant-speed geodesic $$\gamma\colon [0,1]\to X$$.

2. The function $$f$$ is distance-like, in the sense that $$f(x) = \lambda + d\bigl(x, f^{-1}(\lambda)\bigr)$$ for every $$x\in X$$ and every $$\lambda\in (-\infty,f(x)]$$.

### My Question

What, exactly, is the relationship between these two definitions? Are they equivalent? Is the second a generalization of the first? I'd particularly appreciate a reference to a paper that discusses both definitions.

• If you take min of two def1-functions you get a def2-function which might be not a def1-function. (The relation the same as the distance-function to a point and distance-function to a set.) – Anton Petrunin Apr 24 '19 at 4:13
• @AntonPetrunin Taking the min doesn't seem to yield an $\epsilon$-convex function. For example, if $X$ is the real line with $x_0=0$ and the two functions are $f(x)=x$ and $g(x)=-x$, then the minimum of $f$ and $g$ is the negative of the absolute value function, which isn't $\epsilon$-convex for any $\epsilon>0$. – Jim Belk Apr 24 '19 at 8:15
• In the second def you probably assume that $X$ has nonpositive curvature (otherwise it has no sense); in this case any def1-function is convex. – Anton Petrunin Apr 24 '19 at 19:16

I don't know whether this was known before, but Collin Bleak, Francesco Matucci, and I have settled this question in the course of our work on our recent paper [1]. The answer is that any horofunction satisfying Definition 1 satisfies Definition 2, but there exist hyperbolic groups with horofunctions satisfying Definition 2 and not Definition 1.

## Definition 1 implies Definition 2

This is based on the following claim.

Claim. Each of the functions $$f_p$$ is $$2\delta$$-convex.

Proof: Let $$\gamma\colon [0,1]\to X$$ be a constant-speed geodesic of length $$L$$, and let $$a=\gamma(0)$$ and $$b=\gamma(1)$$. Let $$t\in [0,1]$$, and choose geodesics $$[p,a]$$ and $$[p,b]$$ from $$p$$ to $$a$$ and $$b$$, respectively. Since $$X$$ is $$\delta$$-hyperbolic, there exists a point $$q$$ on $$[p,a]\cup [p,b]$$ so that $$d(q,\gamma_t)\leq \delta$$. If $$q\in [p,a]$$ then $$\begin{multline*} d(\gamma_t,p) \leq d(q,p)+\delta = d(a,p) - d(a,q) + \delta \\ \leq d(a,p) - d(a,\gamma_t) + 2\delta = d(a,p)-tL+2\delta \end{multline*}$$ so $$f_p(\gamma_t)\leq f_p(a) - tL+2\delta$$, and a similar statement holds if $$q\in[p,b]$$. Thus $$f_p(\gamma_t) \leq \max\bigl(f_p(a)-tL,f_p(b)-(1-t)L\bigr)+2\delta$$ for all $$t\in[0,1]$$, and it follows that $$f_p(\gamma_t)\leq (1-t)f_p(a)+tf_p(b)+2\delta$$ for all $$t\in[0,1]$$. $$\square$$

Taking a limit, we deduce that any horofunction satisfying Definition (1) is $$2\delta$$-convex. The following claim finishes the proof.

Claim. Any horofunction satisfying Definition (1) is distance-like.

Proof: Let $$\{p_n\}$$ be an unbounded sequence of points in $$X$$ so that $$f_{p_n}$$ converges to a function $$f\colon X\to \mathbb{R}$$ uniformly on compact sets. Let $$x\in X$$ and let $$\lambda\in (-\infty,f(x))$$, so $$f_{p_n}(x) > \lambda$$ for large enough $$n$$. Since the sequence $$\{p_n\}$$ is unbounded, we also know that $$f_{p_n}(p_n)<\lambda$$ for large enough $$n$$. For each such $$n$$, choose a geodesic $$[p_n,x]$$. By the Intermediate Value Theorem, there is a point $$y_n$$ on this geodesic so that $$f_{p_n}(y_n) = \lambda$$. Note then that $$d(x,y_n) = f_{p_n}(x) - \lambda$$ for each $$n$$, so $$d(x,y_n) \to f(x)-\lambda$$ as $$n\to\infty$$. Since $$X$$ is proper, the sequence $$\{y_n\}$$ must have a limit point $$y$$, which satisfies $$f(y) = \lambda$$ and $$d(x,y) = f(x)-\lambda$$. Thus $$d\bigl(x,f^{-1}(\lambda)\bigr) \leq f(x)-\lambda$$, and the opposite inequality follows from the fact that $$f$$ is $$1$$-Lipschitz (since each $$f_p$$ is). $$\square$$

## Definition 2 does not imply Definition 1

As Anton Petrunin suggests, it is sometimes possible to take something like a minimum of functions that satisfy Definition (1) to get a function that satisfies Definition (2) but not Definition (1).

For example, let $$X$$ be the Cayley graph of the group $$G = \langle a,b \mid ab=ba,b^3=1\rangle \cong \mathbb{Z}\times\mathbb{Z}_3$$ where $$x_0$$ is the identity vertex, and note that $$G$$ acts on $$X$$ in a natural way. Let $$T$$ be the triangle of edges connecting $$1$$, $$b$$, and $$b^2$$. Using Definition (1), each point $$p\in T$$ has two associated horofunctions, namely those associated to the sequences $$\{a^n p\}$$ and $$\{a^{-n}p\}$$, and it is not hard to check that these are all of the horofunctions on $$X$$ determined by Definition (1). In particular, the horofunction boundary of $$X$$ by Definition (1) is homeomorphic to the disjoint union of two copies of $$T$$.

However, there are horofunctions for $$X$$ satisfying Definition (2) but not Definition (1). For example, let $$\{g_n\}$$ be the sequence of functions $$g_n(x) = \min\bigl(d(x,a^n),d(x,a^nb),d(x,a^nb^2)\bigr) - n$$ where $$a^n,a^nb,a^nb^2$$ denote the corresponding vertices in $$X$$. Then $$\{g_n\}$$ converges uniformly on compact sets to a function $$g\colon X\to\mathbb{R}$$ which satisfies Definition (2) but not Definition (1). In particular, $$g(1)=g(b)=g(b^2)=0$$, but this is not true for any horofunction satisfying Definition (1). More generally, for any values $$u,v\in [-1,1]$$ satisfying $$|u-v|\leq 1$$, there are exactly two Definition (2) horofunctions on $$X$$ that satisfy $$g(1)=0$$, $$g(b)=u$$, and $$g(b^2)=v$$. The set of all such $$u$$ and $$v$$ is a closed hexagonal region in $$\mathbb{R}^2$$, and the set $$\Phi$$ of horofunctions satisfying Definition (2) is the union of two disjoint copies of this region.

Note that $$G\times \mathbb{Z}_3$$ will have similar behavior for any hyperbolic group $$G$$. In particular, there exist non-elementary hyperbolic groups for which Definition (1) and Definition (2) are not equivalent. Note also that the function $$g$$ defined above takes integer values on vertices, so it corresponds to a point in the space $$\Phi_0$$ of integral coycles defined by Coornaert and Papadopoulos.

• Gromov-hyperbolic spaces are coarse objects, so the right thing to ask is not that 2 implies 1 but that 2 implies 1 up to a uniformly bounded error. This is indeed the case, and it follows from Proposition 6.1 in Coornaert-Papadopoulos. There are other variations on the same definition, see for instance, the paper by Bowditch on relatively hyperbolic groups (he uses the negative of the usual horofunctions) and in Buyalo-Schroeder "Elements of Asymptotic Geometry". – Misha Sep 20 '19 at 15:39