A sum-bracket theorem for simple Lie algebras

Let $\mathfrak{g}$ be an algebra over $K$ with a bilinear operation $[\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$ not necessarily associative. For $A\subseteq\mathfrak{g}$, let $A^{k}$ be the set of elements of $\mathfrak{g}$ written combining $k$ elements of $A$ via $+$ and $[\cdot,\cdot]$. We show a "sum-bracket theorem" for simple Lie algebras over $K$ of the form $\mathfrak{g}=\mathfrak{sl}_{n},\mathfrak{so}_{n},\mathfrak{sp}_{2n},\mathfrak{e}_{6},\mathfrak{e}_{7},\mathfrak{e}_{8},\mathfrak{f}_{4},\mathfrak{g}_{2}$: namely, if $\mathrm{char}(K)$ is not too small, we have exponential growth of the form $|A^{k}|\geq|A|^{1+\varepsilon}$ for all generating symmetric sets $A$ whose size is not too close to that of a subfield of $K$. Over $\mathbb{F}_{p}$ in particular, we have a diameter bound in line with analogous bounds for groups of Lie type, matching the best qualitative estimates of [BDH21]. As an independent intermediate result, we prove also a dimensional estimate for linear affine subspaces $V$ of $\mathfrak{g}$, i.e. an estimate of the form $|A\cap V|\leq|A^{k}|^{\dim(V)/\dim(\mathfrak{g})}$. This estimate is valid for all simple algebras. Moreover, $k$ is especially small for a large class of algebras: these include associative algebras, Lie algebras, Lie superalgebras, and Mal'cev algebras with $\mathrm{char}(K)\neq 2$.Comment: 36 page

The diameter of products of finite simple groups

Following partially a suggestion by Pyber, we prove that the diameter of a product of non-abelian finite simple groups is bounded linearly by the maximum diameter of its factors. For completeness, we include the case of abelian factors and give explicit constants in all bounds.Comment: 8 page

Topological full groups of minimal subshifts and quantifying local embeddings into finite groups

We investigate quantitative aspects of the LEF property for subgroups of the topological full group $[[ \sigma ]]$ of a two-sided minimal subshift over a finite alphabet, measured via the LEF growth function. We show that the LEF growth of $[[ \sigma ]]^{\prime}$ may be bounded from above and below in terms of the recurrence function and the complexity function of the subshift, respectively. As an application, we construct groups of previously unseen LEF growth types, and exhibit a continuum of finitely generated LEF groups which may be distinguished from one another by their LEF growth.Comment: 20 pages, comments welcome