I'm interested in whether the finitely-generated discrete Heisenberg group admits a notion of "convex set". Below a formalization of what I need from the convex sets, in particular they should all be finite, should be arbitrarily large, the family should be translation-invariant, closed under intersections and satisfy anti-exchange. They should also have the (non-standard?) property that $g$ is always in the closure of $\{gh, gh^{-1}\}$.
Say a convex geometry $\mathcal{C} \subset 2^X$ is $\{\tau(S) \;|\; S \subset X \mbox{ finite}\}$ where $X$ is a set, and $\tau$ is closure operator which takes finite sets to finite sets, and additionally satisfies the anti-exchange axiom $$ a, b \notin \tau(A) \implies (a \notin \tau(A \cup \{b\}) \vee b \notin \tau(A \cup \{a\})). $$ (I took this from Wikipedia's page on antimatroids, it seems to be standard.)
If $\mathcal{C}$ is a convex geometry on a group $G$, we say it is translation-invariant (for the left action of the group on itself) if $C \in \mathcal{C} \implies \forall g \in G: gC \in \mathcal{C}$. Say a convex geometry on $G$ is midpointed if it satisfies $g \in \tau(\{gh, gh^{-1}\})$ for all $g, h \in G$.
(I wrote a proof sketch that a translation-invariant convex geometry is midpointed if and only if for all $C, D \in \mathcal{C}$ and for all $a \in C$ such that $C \setminus \{a\} \in \mathcal{C}$, there is a at most one $g \in G$ such that $gD \subset C$. This is the actual property that interests me, but it's perhaps less pretty on the first sight so I opted for the above.)
The sets $C \cap \mathbb{Z}^d$ where $C$ ranges over compact convex subsets of $\mathbb{R}^d$ form a midpointed translation-invariant convex geometry on the group $\mathbb{Z}^d$. If $G$ is not torsion-free, then it does not admit any midpointed convex geometry (an involution forces $1_G$ and vice versa, so anti-exchange fails, and there's a similar argument for other $\mathbb{Z}_p$). Neither does the abelian group $\mathbb{Z}[1/2]$ (because $0, 1$ necessarily has infinite closure), nor the finitely-generated metabelian group $\mathbb{Z}[1/2] \rtimes \mathbb{Z}$ where $\mathbb{Z}$ acts by multiplication by $2$.
Question 1. Does the discrete Heisenberg group of integer matrices of the form $\left( \begin{smallmatrix} 1 & a & c \\ 0 & 1 & c \\ 0 & 0 & 1 \end{smallmatrix}\right)$ admit a midpointed translation-invariant convex geometry?
and more generally
Question 2. Which strongly polycyclic groups admit a midpointed translation-invariant convex geometry?
I'm also interested in the following, though I have no direct use for it, and I imagine it's easier to find information about.
Question 3. What other midpointed translation-invariant convex geometries does $\mathbb{Z}^d$ admit?