Growth Expansion and Mixing in Groups and Graphs

Definitions and mathematical statement

The record collects several related growth and expansion questions, without specifying a single quantitative conjecture.

1. For a graph $\Gamma$ and a starting vertex $v$, determine the number of vertices reachable in time $t$. With unit-time edge traversal this is $|\{w:d_\Gamma(v,w)\leq t\}|$; other travel rules require specification.
2. For a generating subset $A$ of an infinite group $G$, retain the word-product counting function \[ B_A(t)=\left|\bigcup_{j=0}^{t}A^j\right|,\qquad A^0=\{1\}, \] and determine its growth. Inverses are not silently adjoined to $A$; finite generation and symmetry are not explicitly stated in the extract.
3. For the finite-group random walk generated by $A$, determine the rate of approach to uniform measure. Given a step probability $\mu$ supported on $A$, a standard measure of mixing is \[ \frac12\sum_{g\in G}|\mu^{*t}(g)-|G|^{-1}|. \] The step law, metric and periodicity convention are absent. Generation alone need not give convergence without aperiodicity.
4. Determine lower bounds for the gap between the two largest adjacency eigenvalues. In the undirected regular Cayley setting the adjacency operator is $Tf(g)=\sum_{a\in A}f(ga)$ and the requested gap is $\lambda_1-\lambda_2$. Real ordered eigenvalues require an appropriate self-adjoint convention, such as inverse-closed $A$. Normalized gap and absolute spectral gap are different quantities; the latter also detects periodicity through negative eigenvalues.
5. For finite nonempty $A$ in an abelian group, determine how $|A+A|$ compares with $|A|$; for finite nonempty $A$ in a general group, determine how $|A^3|$ compares with $|A|$, where $A^3=\{abc:a,b,c\in A\}$, and understand the structure responsible for growth or its failure.

The graph and group classes, size regimes, generation/symmetry assumptions, random-walk law and desired strengths of the growth or gap estimates are not supplied. Consequently the broad collection is retained rather than replaced by a uniform expansion theorem for all groups or a specific product-growth exponent. The infinite-group growth and finite-group mixing questions retain their different domains.

Short English statement

Study graph-ball and word growth, finite-group mixing and spectral gaps, and sumset/triple-product growth. The source presents a collection of research directions whose quantitative targets and hypotheses are not specified, with product-growth results in important special families.

Sources

• Supplied report: Analytic Number Theory (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/14, pp. 669--746; proposed year 2008. https://doi.org/10.4171/owr/2008/14.
• Supplied Helfgott $\operatorname{SL}_3$ growth theorem reference: https://doi.org/10.4171/JEMS/267.
• Supplied reference labelled "growth of small $\operatorname{SL}_n$ generating sets": https://doi.org/10.1093/imrn/rnq244.
• Local UnsolvedMath registry, permanent ID 30000935, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note says the full collection was retained with notation repaired. The supplied 21 August 2026 triage labels it partially solved, reporting strong $\operatorname{SL}_3$ product growth and related diameter/expansion consequences, with extensions to other linear and finite simple groups, but not a complete resolution of all broad questions. This dated assessment is not independently verified here.