Embeddings of gromov hyperbolic spaces

The Gromov boundary has several important properties. One of the most frequently used properties in group theory is the following: if a group \(G\) acts geometrically on a \(\delta\) -hyperbolic space, then \(G\) is hyperbolic group and \(G\) and \(X\) have homeomorphic Gromov boundaries. One of the most important properties is that it is a quasi-isometry invariant; that is, if two hyperbolic metric spaces are quasi-isometric, then the quasi-isometry between them gives a homeomorphism between their boundaries. This is important because homeomorphisms of compact spaces are much easier to understand than quasi-isometries of spaces.

Suppose that \(f:X\to Y\) is a power quasisymmetry of bounded metric spaces \(X, Y\). Then \(\hat{f}:Con(X)\to Con(Y)\) is a rough quasi-isometry. If \(f\) is a snowflake or bilipschitz map, then \(\hat{f}\) is a rough similarity or a rough isometry, respectively.

Suppose \(X\) is a visual Gromov Hyperbolic Metric Space. Then \(X\) and \(Con(\partial X)\) are rougly similar.

• It can be shown that a metric space has finite Assouad dimension if and only if it is doubling.
• This conclusion is proved in On the upper regularity dimensions of measures
• Actually the real proof can only be found in: J. C. Robinson. Dimensions, Embeddings, and Attractors, Cambridge University Press, Cambridge, (2011).

Dimensions, embeddings, and attractors

• One of the most powerful tools for providing lower bounds for the Assouad dimension of a set is the well-known result of Mackay and Tyson concerning weak tangents: Conformal dimension. Theory and application
• Conformal dimension: theory and application

从metric角度的性质导出（欧式空间）几何性质

• Gromov Hyperbolic Geodesic Metric Space
  
  1. bounded growth at some scale are all complete simply-connected Riemannian \(n\) -manifolds \(X\) with sectional curvature \(\kappa\) satisfying \(-b^2\le \kappa\le -a^2<0\).
     
  2. Complex Hyperbolic Space
• \(X\) 的一个较弱的条件可以导出边界 \(\partial X\) 一个较强的结论
• 用了很硬核的分析技巧

\[n\le N^m = N^{1+\frac{R*}{R-r}}\le C(R,r,N,\delta)\left(\frac{\beta}{\alpha}\right)^{C(R,r,N,\delta)}\]

plug in \(R^*=log(\frac{\beta}{\alpha})+C(\delta)\)

\[N^{1+\frac{R*}{R-r}} = N^{^{}1+\frac{log(\frac{\beta}{\alpha})+C(\delta)}{R-r}}\]

\[=N^{1+\frac{C(\delta)}{R-r}}\cdot N^{log\left(\left(\frac{\beta}{\alpha}\right)^{(R-r)^{-1}}\right)}\]

To compute constant \(K\) and the exponent \(s\):

let \[\left(\frac{\beta}{\alpha}\right)^s=N^{log\left(\left(\frac{\beta}{\alpha}\right)^{(R-r)^{-1}}\right)}\]

\[s\cdot log\left(\frac{\beta}{\alpha}\right)=\frac{logN}{R-r}\cdot log\left(\frac{\beta}{\alpha}\right)\]

then we can get \[s=\frac{logN}{R-r}\]

the upper bound is further simplified to \[N^{1+\frac{C(\delta)}{R-r}}\cdot \left(\frac{\beta}{\alpha}\right)^{\frac{logN}{R-r}}\]

recall the definition of Assouad Dimension

\[s=\frac{logN}{R-r}, K=N^{1+\frac{C(\delta)}{R-r}}\]

while in Algorithms on negatively curved spaces, the condition stated in local geometry.

Luckily, a straightforward corollary can be employed to perform transformation between these two definitions:

\[B\left(mR-(m-1)r\right)\sim N^m \cdot B(r)\]

we want to deduce the equivalent expression in local geometry:

\[B(R_0)\sim \lambda_0 \cdot B(\frac{R_0}{2})\]

let \(r=\frac{R_0}{2}\)

\[mR-(m-1)r=R_0\]

\[m(R-r)+r=R_0\]

\[m=\frac{R_0-r}{R-r}\]

\[m=\frac{r}{R-r}\]

then \(\lambda_0 = N^m\)

\[log\lambda_0 = m logN\]

\[log\lambda_0 = \frac{rlogN}{R-r}\]

combined with the result: \(Dim_A(\partial X)=O\left(\frac{logN}{R-r}\right)\)

then \(Dim_A(\partial X)=O\left(\frac{2log\lambda_0}{R_0}\right)\)

according to Lemma 9.4 Finite Assouad Dimenson VS Doubling:

\[Dim_A(\partial X)=log_2 K\]

wherein \(K\) denotes the doubling constant.

then, combined all above together:

\[K=\lambda(\partial X)=2^{Dim_A(\partial X)}=2^{O\left(\frac{2log\lambda_0}{R_0}\right)}\]

compute the exponent again:

let \(2^{O\left(\frac{2log\lambda_0}{R_0}\right)}=\lambda_0^x\)

\[\frac{2log2\cdot log\lambda_0}{R_0}=xlog\lambda_0\]

\[x=\frac{2log2}{R_0}\]

finally, we get:

\[\lambda(\partial X) =\lambda_0^{O\left(1/R_0 \right)}\]

if \(A\subset\mathbb{H}^n\cup \partial\mathbb{H}^n\), let \(hull(A)\subset\mathbb{H}^n\) be the intersection of all closed half-spaces \(H\subset \mathbb{H}^n\) such that \(A\subset H\cup\partial H\).

(1) Let \(A\subset \mathbb{H}^n\). For each \(p\in hull(A)\), there exists a point \(q\in\mathbb{H}^n\) that lies on a geodesic segment with endpoints in \(A\) and satisfies \(d_h(p,q)\le O(1)\).
(2) Let \(A\subset \partial\mathbb{H}^n\) be a set with more that one point, and fix \(o\in hull(A)\). For each \(p\in hull(A)\), there exists a point \(q\in\mathbb{H}^n\) that lies on a geodesic ray from \(o\) to some point in \(A\) and satisfies \(d_h(p,q)\le O(1)\).
(3)Let \(Z\subset \mathbb{R}^{n-1}\subset\partial\mathbb{H}^n\) be a compact set containing more than one point. Then \(hull(Z)\sim Con(Z)\).

Let \(X\) be a Gromov hyperbolic geodesic metric space with bounded growth at some scale. Then there exists an integer \(n\) such that \(X\) is roughly similar to a convex subset of hyperbolic \(n\) -space \(\mathbb{H}^n\).

• \(X\) is visual, and so \(X\sim Con((\partial X, d))\) by Theorem 8.2. \(d\) is some fixed metric in the canonical gauge on \(\partial X\).
  
• By Theorem 7.4 applied to snowflake maps, \(Con((\partial X, d))\sim Con((\partial X, d^{1/2}))\).
• Theorem 9.2 shows that \(\partial X\) has finite Assouad dimension.
• By Theorem 9.1, \((\partial X, d^{1/2})\) admits a bilipschitz embedding into \(\mathbb{R}^{n-1}\) for sufficiently large \(n\).
• By Theorem 7.4 for bilipschitz maps, \(Con((\partial X, d^{1/2}))\sim Con(Z)\).
• Proposition 10.1 shows that \(Con(Z)\sim hull(Z)\). Therefore, \(X\sim hull(Z)\subset\mathbb{H}^n\), proving the assertion in this case.