6 problems

Pairing-matrix conjecture. If the generators $a_1,\ldots,a_n$ and $b_1,\ldots,b_m$ of $\Gamma_C$ are distinct, then $\Gamma_C$ contains a nontrivial element of finite order. This i…

Bellaterra expander conjecture. The Bellaterra graphs $\{B_i\}_{i=1}^\infty$ are a family of two-sided expanders. The authors' computations suggest an eigenvalue gap of roughly…

Let a $p$-letter automaton generate a semigroup, let its growth series be the generating function recording the number of elements of each length, and let its dual be the automaton…

Let a reversible Mealy automaton be a Mealy automaton whose transition maps are permutations, and let the semigroup generated by an automaton be the semigroup generated by its stat…

Let $p>2$ and $q>2$ when considering the displayed family, together with the specializations for $p=2$ or $q=2$. Let $\langle\aut{M}_{p,q}\rangle$ denote the group generated by the…

Let $\mathcal A$ be an invertible Mealy automaton with an arbitrary stateset. Free-semigroup extension conjecture. Proposition should extend to $\mathcal A$, so that its conclusion…