Let $\Gamma = \langle a,b,c \ | \ c=aba^{-1}b^{-1}, \ ac=ca, \ bc = cb \rangle$ be the discrete Heisenberg group.
Let $\ell: \Gamma \to \mathbb{N} $ be the word length on $\Gamma$. This group has a polynomial growth rate of order $4$.
Let $H$ be the separable Hilbert space $l^2(\Gamma)$. Let $A$ be the ${\rm C}^{\star}$-algebra ${\rm C}^{\star}_r(\Gamma)$.
Let $D$ be the (unbounded) Dirac operator on $H$, defined as an extension of the map $ \gamma \mapsto \ell(\gamma)\gamma$.
Question: Is there a non-integer in the dimension spectrum of the spectral triple $(A,H,D)$?
Remark: The intuitive reason why we expect the existence of a non-integer in this dimension spectrum, is that the Heisenberg group admits $3$ generators, but a polynomial growth rate of order $4$, so something should happen strictly between $3$ and $4$. Moreover, non-integers appear in some results about some generalized Heisenberg groups (see this paper of Michael Stoll).
