Some Questions — Question 29

Definitions and mathematical statement

Let $\Gamma$ be an infinite group generated by a finite symmetric set $S$ not containing the identity. Its word length is $|g|_S=d(1,g)$ in the unit-edge Cayley graph. An element $g$ is a dead end relative to this generating set if \[ |gs|_S\leq |g|_S\qquad\text{for every }s\in S. \] Thus no edge from $g$ continues away from the identity to the next word-length level. This is the depth-one dead-end condition; larger notions of dead-end depth are not used.

For the word ball $B_R=\{g:|g|_S\leq R\}$, define \[ D_S(R)=\frac{|\{g\in B_R:g\text{ is a dead end}\}|}{|B_R|}. \] Does $D_S(R)\to0$ as $R\to\infty$ for every infinite finitely generated group and every such finite generating set?

The averaging is over all vertices in the ball, rather than only over its outer sphere or a random-walk distribution. The generating set is fixed throughout the limit and is part of the universally quantified input. The problem does not assert that dead ends are absent, that their depths are bounded, or that their number is finite; infinitely many exceptional vertices are consistent with zero density. Conversely, a counterexample need only have positive limsup density along some radii, so no assumed existence of a density limit is built into the question. Distinct group elements, rather than geodesic words representing them, are counted.

Short English statement

Does the proportion of dead ends inside growing word balls tend to zero for every infinite Cayley graph?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1200029 (AMR-011-0029), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Abert - Some questions (pdf) (2010), Question 29. Original source indexed by AMR-011-0029. https://www.renyi.hu/~abert/questions.pdf.