Suppose I have a uniformly Holder continuous plane field $H$ on $\mathbb{R}^{3}$. I will assume that this plane field $H$ has many special properties, all of which are completely unreasonable to expect for a general such plane field $H$. These are summarized below,

(1) There is a constant $C > 0$ such that for every $x, y \in \mathbb{R}^{3}$ with $d(x,y) \leq 1$ ($d$ denoting Euclidean distance) there is a $C^1$ path $\gamma$ tangent to $H$ connecting $x$ to $y$ of length $ \leq C$.

(2) There is a smooth uniformly expanding map $f: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ which preserves $H$ (so we are free to dilate paths tangent to $H$).

(3) There is a smooth $f$-invariant line field $V$ transverse to $H$.

(4) Let $\rho$ be the path metric defined by taking $\rho(x,y)$ to be the infimal length of paths connecting $x$ to $y$ which are tangent to $H$. There is a locally finite Borel measure $\mu$ on $\mathbb{R}^{3}$ such that the metric measure space $(\mathbb{R}^{3}, \rho, \mu)$ is Ahlfors $Q$-regular for some $Q > 0$.

There is one obvious way to construct such a plane field with the metric properties above, which is to take a $C^{1+Holder}$ diffeomorphic image of the 3-d Heisenberg group and take the image of the Carnot-Caratheodory metric defined by the horizontal contact distribution.

My question: Through considering a different problem I encountered metric measure spaces of the type described above which are not (locally) quasisymmetric to the 3-dimensional Heisenberg group. I'm wondering if either (a): anyone has encountered any spaces like this before, or (b): conditions (1)-(4) impose any new nonobvious properties on the space. The most basic question that can be asked is whether the set of all rectifiable curves in this space $(\mathbb{R}^{3}, \rho, \mu)$ has positive $Q$-modulus, but this probably (maybe?) can't be shown without additional hypotheses.

This may be too open-ended, but part of my problem is that I don't even know what questions one would necessarily want to ask and have answered about such spaces as well.

Edit: Actually there is a second natural question here, which is whether it's possible that we could have $Q < 4$.

Edit 2: I realized I could further point out the abundance/oddness of these spaces by the fact that they actually exist in 1-parameter families: for each $t \in (-\varepsilon,\varepsilon)$, $\varepsilon > 0$ small, there exists an $H_{t}$, $\rho_{t}$, $\mu_{t}$, $Q_{t}$ etc. continuously depending on $t$ such that at $t = 0$ you get the Heisenberg group with its standard Carnot-Caratheodory metric and none of these spaces are locally quasisymmetrically equivalent for distinct values of $t$.


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