Let $M$ be a Riemmanian manifold such that for all $x\in M$ and large $R>0$, $$Vol(B(x, R))\leq \exp(R).$$ Let $p\in M$ and $d$ be the geodesic distance. Suppose that for large enough $r$ and all $x, y\in \partial B_{r}(p)$, $$d(x, y)\leq c\log r.$$ Is it then true that there exist constants $N>0$ and $C>0$ such that for all large enough $r$ we have $$Vol(B(p, r))\leq Cr^{N}?$$
Thanks in advance for your help!
Yes.
In more details, we can pick $D>1$ be so that $\text{Vol}(B_R(x))<e^R$ for ALL $x$ whenever $R\geq D$. Take the first integer $L>1$ so that $L>e^{D/c}$, then $c\log L>D$. For any $s\geq L$, note $2c\log s-c\log s=c\log s\geq c\log L>D>1$.
Take $x\in \partial B_s(p)$, observe $$ \{z\,|\, s\leq d(z, p)\leq s+1\}\subset B_{2c\log s}(x); $$
in fact, for any $z$ with $s\leq d(z, p)\leq s+1$, take a minimal geodesic $\gamma$ from $p$ to $z$, then $z=\gamma(t)$ with $t=d(z, p)\in [s, s+1]$, so there is a point $y=\gamma(s)\in \partial B_s(p)$ so that $d(y, z)=d(\gamma(t), \gamma(s))\leq 1$. Now $d(x, y)\leq c\log s$, so $$ d(z, x)\leq d(z, y)+d(y,x)\leq 1+c\log s\leq 2c\log s. $$
In particular, let $N=2c+1$, we see $$ \text{Vol}(\{z\,|\, s\leq d(z, p)\leq s+1\}) \leq \text{Vol}(B_{2c\log s}(x))\leq e^{2c\log s}=s^{2c}=s^{N-1}. $$
Let $C_1$ be so that $\text{Vol}(B_L(p))=C_1L^N$. Now for any $r>L$, let $k$ be $L+k\leq r<L+k+1$, so $$ \begin{aligned} \text{Vol}(B_r(p))\leq &\text{Vol}(B_L(p))+\text{Vol}(\{z\,|\, L\leq d(z, p)\leq L+1\})\\ &+\text{Vol}(\{z\,|\, L+1\leq d(z, p)\leq L+2\})+...\\ &+\text{Vol}(\{z\,|\, L+k\leq d(z, p)\leq L+k+1\})\\ \leq & C_1L^N+L^{N-1}+(L+1)^{n-1}+...+(L+k)^{N-1}\\ \leq & C_1L^N+(k+1)(L+k)^{N-1}\\ \leq & C_1(L+k)^N+(L+k)^{N}=(C_1+1)(L+k)^N\leq (C_1+1)r^N. \end{aligned} $$