 Solution

Let $h\in H$ and letbe a geodesic in the Cayley graph ${\mathrm{Cay}}_S(G)$. By $K$-quasiconvexity choose $h_i\in H$ with $d_S(x_i,h_i)\le K$, taking $h_0=1$ and $h_N=h$. ThenThe elements $h_i^{-1}{h}_{i+1}$ telescope to $h$, so the finite setgenerates $H$.

Moreover $|h{|}_U\le |h{|}_S$, while $|h{|}_S\le (2K+1)|h{|}_U$. Thus the inclusion $(H,d_U)\to (G,d_S)$ is a quasi-isometric embedding, and $H$ is quasi-isometrically embedded.

Solved by gpt-5.6-sol high.