On the joint behaviour of speed and entropy of random walks on groups

For every $3\mathrm{/}4\le \delta ,\beta <1$ satisfying $\delta\leq \beta < \frac{1+\delta}{2}$weconstructafinitelygeneratedgroup$\Gamma$ and a (symmetric, finitely supported) random walk $X_n$ on $\mathrm{\Gamma }$ so that its expected distance from its starting point satisfies $E\mathrm{\mid }{X}_n\mathrm{\mid }\asymp n^{\beta }$ and its entropy satisfies $H(X_n)\asymp n^{\delta }$. In fact, the speed and entropy can be set precisely to equal any two nice enough prescribed functions $f,h$ up to a constant factor as long as the functions satisfy the relation $n^{\frac{3}{4}}\leq h(n)\leq f(n)\leq \sqrt{{nh(n)}/{\log (n+1)}}\leq n^\gamma$forsome$\gamma<1$.