Let me give an explicit example for the Poincare disk in dimension 2. In the disk model, just take the regular 4g-gon centered on the origin. Identifying opposite sides, we get a compact surface of genus g. The polygon is a fundamental domain for the group $\Gamma$ generated by $b_1...b_{2g-1}$ with

$b_k=\pmatrix{cosh(a) & e^{ik\pi\over 2g}sinh(a)\cr e^{-ik\pi\over 2g}sinh(a)& cosh(a)}$

where a is defined by $cosh(a)=1/tan(\pi/4g)$. The value of a is in fact equal to the radius of the maximal disk contained in the fundamental domain. This is also the distance from any side of the 4g-gon to the origin. It goes to infinity with g, so we are done.

The Gauss-Bonnet formula tells us that surfaces of a given genus have a bounded volume, hence cannot contains arbitrarily large embedded balls. Hope that helps.

Let me give an explicit example for the Poincare disk in dimension 2. In the disk model, just take the regular $4g$-gon centered on the origin. Identifying opposite sides, we get a compact surface of genus g. The polygon is a fundamental domain for the group $\Gamma$ generated by $b_1...b_{2g-1}$ with

$b_k=\pmatrix{\cosh(a) & e^{ik\pi\over 2g}\sinh(a)\cr e^{-ik\pi\over 2g}\sinh(a)& \cosh(a)}$

where a is defined by $\cosh(a)=1/\tan(\pi/4g)$. The value of a is in fact equal to the radius of the maximal disk contained in the fundamental domain. This is also the distance from any side of the 4g-gon to the origin. It goes to infinity with g, so we are done.

The Gauss-Bonnet formula tells us that surfaces of a given genus have a bounded volume, hence cannot contains arbitrarily large embedded balls. Hope that helps.