Skip to main content
MathOverflow

Return to Question

fixed typ
Source Link
Jim Belk
Jim Belk
  • 8.5k
  • 1
  • 42
  • 52

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 $\gamma\colon [0,1]\to X$.
  1. 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.

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 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 $\gamma\colon [0,1]\to X$.
  1. 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.

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 $\gamma\colon [0,1]\to X$.
  1. 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.

clarified lambda
Source Link
YCor
YCor
  • 63.9k
  • 5
  • 187
  • 285

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 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 $\gamma\colon [0,1]\to X$.
  1. 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\leq f(x)$$\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.

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 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 $\gamma\colon [0,1]\to X$.
  1. 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 $\lambda\leq f(x)$.

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.

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 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 $\gamma\colon [0,1]\to X$.
  1. 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.

Source Link
Jim Belk
Jim Belk
  • 8.5k
  • 1
  • 42
  • 52

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 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 $\gamma\colon [0,1]\to X$.
  1. 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 $\lambda\leq f(x)$.

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.