We try to motivate our question. We have a certain logical/operational structure that has an emergent physical interpretation. We are giving this structure a geometric setting via quasi-isometries. We wonder what restrictions the structure places on the possible geometric settings. We know that the set of solutions to our problem is not empty (see preprint http://arxiv.org/abs/1306.1934). The question seems relevant in connecting a new topic in physics (emergent space-time from discrete Planck scale) with a new topic in math: geometric-group-theory. Quantum cellular automata seem to "physically" achieve the quasi-isometric embedding of geometric-group theory. If you are interested in the physics, look at the link above.

In the following $\mathbb{C}$ denotes the set of complex numbers, $\mathbb{M}_n(\mathbb{C})$ the $n\times n$ complex matrices, $A^\dagger$ the adjoint of matrix $A$.

Let's call *Isotropic Quantum Cellular Automaton* (IQCA) the set $$[G,S_+,L,T,U,
[A_h]_{h\in S}]$$

where

- $G$ is a (countable generally infinite) group that is finitely generated
- $S_+$ is a finite set of generators for a $G$ (we will denote by $S_-:=S_+^{-1}$, and we let $S=S_+\sqcup S_-\sqcup\{e\}$, with $e$ the identity of $G$)
- $L$ is a finite subgroup of the automorphism group of $G$. The action of $L$ preserves $S_+$ as a set, and is transitive on $S_+$
- $T$ is a faithful irreducible unitary representation of $G$ on the separable Hilbert space ${\ell}^2(G)$
- $V$ is a faithful irreducible unitary representation of $L$ on the Hilbert space
$\mathbb{C}^n$, for some $n<\infty$ (for $n=1$ the IQCA is called
*trivial*) - $\{A_h\}_{h\in S}\subseteq\mathbb{M}_n(\mathbb{C})$ (called
*transition matrices*) - the following operator is unitary over ${\ell}^2(G)\otimes\mathbb{C}^n$: \begin{equation} A=\sum_{h\in S}T(h)\otimes A_h \end{equation} and is covariant under $V(L)$, namely \begin{equation} \sum_{h\in S}T(h)\otimes A_h=\sum_{h\in S}T(lh)\otimes V(l)A_hV(l)^\dagger,\quad\forall l\in L. \end{equation}

We are interested only in nontrivial IQCA. We say that the IQCA is quasi-isometrically embeddable (qie) in a $d$-dimensional Riemannian manifold $M^d$ if the Cayley graph $\Gamma(G,S_+)$ has a one-to-one quasi-isometry to $M^d$ (we use the word metric on $\Gamma(G,S_+)$).

**Questions:**

- Are there necessary conditions so that for a given $G,S_+,L$ there exists a non trivial IQCA? [Simpler: replace $M^d$ by $\mathbb{R^3}$.]
- Are there sufficient conditions so that for a given $G,S_+,L$ there exists a non trivial IQCA? [Simpler: replace $M^d$ by $\mathbb{R^3}$.]
- Do there exist nontrivial IQCA qie in a $d$-dimensional manifold with $d=2,3$ and constant nonzero curvature?