Grigorchuk's modified Milnor problem

A group is finitely presented if it admits a finite presentation, and its algebraic entropy measures the exponential growth rate of balls in a finite generating set. Vanishing algebraic entropy means that this growth rate is zero; a group has at most polynomial growth if the sizes of its finite generating-set balls are bounded above by a polynomial.

Grigorchuk's modified Milnor problem. A finitely presented group with vanishing algebraic entropy has at most polynomial growth.

This is a modified form of Milnor's problem on whether vanishing algebraic entropy forces polynomial growth. The source explains that Grigorchuk's original counterexample to Milnor's problem is finitely generated but not finitely presented, and presents the finitely presented version as a conjecture; the paper's main theorem proves the geometric conjecture above rather than stating a resolution of this group-theoretic problem.