On the fundamental length of quantum geometry and the black hole entropy

The geometric operators of area, volume, and length, depend on a fundamental length l of quantum geometry which is a priori arbitrary rather than equal to the Planck length l_P. The fundamental length l and the Immirzi parameter $\gamma$ determine each other. With any l the entropy formula is rendered most naturally in units of the length gap sqrt{{sqrt 3}/2} (sqrt{gamma} l). Independently of the choice of l, the black hole entropy derived from quantum geometry in the limit of classical geometry is completely consistent with the Bekenstein-Hawking form. The extremal limit of 1-puncture states of the quantum surface geometry corresponds rather to an extremal string than to a classical horizon.