Here is a very ad hoc proof that $Rips_2(\mathbb{Z}^2)$ is contractible, which occurred to me at some point in discussions with Brendan Mallery, about a year after I wrote the paper that Ian Agol linked to above.

Consider the maximal simplex $\{(n,m),(n+1,m),(n-1,m),(n,m+1),(n,m-1)\}$. This has $\{(n+1,m),(n-1,m),(n,m+1),(n,m-1)\}$ as a free face (meaning it is the only simplex properly containing that face), so we can delete the simplex and this free face without changing the homotopy type of the complex. Do this for every $n$ and $m$. Within the resulting subcomplex, the (now maximal) simplex $\{(n,m),(n+1,m),(n-1,m),(n,m+1)\}$ has $\{(n+1,m),(n-1,m),(n,m+1)\}$ as a free face, so these can be deleted without changing the homotopy type. Similarly we can "pair up" $\{(n,m),(n+1,m),(n-1,m),(n,m-1)\}$ with $\{(n+1,m),(n-1,m),(n,m-1)\}$ and delete them, and then also the version where you use $n+1,m+1,m-1$ and the version where you use $n-1,m+1,m-1$. Finally, pair up $\{(n,m),(n+1,m),(n-1,m)\}$ with $\{(n+1,m),(n-1,m)\}$ and $\{(n,m),(n,m+1),(n,m-1)\}$ with $\{(n,m+1),(n,m-1)\}$.

After deleting all these pairs of simplices for all $m$ and $n$, the homotopy type has never changed and now we have removed all simplices containing edges of the form $\{(n+1,m),(n-1,m)\}$ or $\{(n,m+1),(n,m-1)\}$. The subcomplex of what's left over (assuming I didn't forget any cases) is a "quilt of tetrahedra", that is, the flag complex of the graph obtained from the standard Cayley graph of $\mathbb{Z}^2$ by adding in every edge of the form $\{(n,m),(n+1,m+1)\}$ and $\{(n,m),(n+1),(m-1)\}$. This is visibly contractible, so $Rips_2(\mathbb{Z}^2)$ is contractible. (By the way, this remove-lots-of-pairs-of-simplices thing can all be phrased using Forman's discrete Morse theory.)

Like I said, this is very ad hoc, and it seems like it would be a big mess to try and generalize to $Rips_k(\mathbb{Z}^2)$ for $k>2$ (I guess your actual question was more about "large radius"), but at least here is a concrete example of a non-hyperbolic group with a contractible Rips complex using a word metric. In general, surprisingly little is known about Rips complexes of groups using word metrics!

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