In [5], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions $\mathcal{F}$ in a domain $\mathit{D}\subset \mathbb{C}$, and for a positive constant ε, if for each $\mathit{f}\in \mathcal{F}$ there exist meromorphic functions ${\mathit{a}}_{\mathit{f}},{\mathit{b}}_{\mathit{f}},{\mathit{c}}_{\mathit{f}}$ such that f omits ${\mathit{a}}_{\mathit{f}},{\mathit{b}}_{\mathit{f}},{\mathit{c}}_{\mathit{f}}$ in D and

$$min\{\mathit{\rho }({\mathit{a}}_{\mathit{f}}(\mathit{z}),{\mathit{b}}_{\mathit{f}}(\mathit{z})),\mathit{\rho }({\mathit{b}}_{\mathit{f}}(\mathit{z}),{\mathit{c}}_{\mathit{f}}(\mathit{z})),\mathit{\rho }({\mathit{c}}_{\mathit{f}}(\mathit{z}),{\mathit{a}}_{\mathit{f}}(\mathit{z}))\}\ge \mathit{\epsilon }$$

for all $\mathit{z}\in \mathit{D}$, then $\mathcal{F}$ is normal in D. Here, ρ is the spherical metric in $\hat{\mathbb{C}}$. In this paper, we establish the high-dimensional versions for the above result and for the following well-known result of Lappan: A meromorphic function f in the unit disc $△:=\{\mathit{z}\in \mathbb{C}:|\mathit{z}|<1\}$ is normal if there are five distinct values ${\mathit{a}}_1,\dots ,{\mathit{a}}_5$ such that

$$sup\{(1-|\mathit{z}{|}^2)\frac{|{\mathit{f}}^{\prime }(\mathit{z})|}{1+|\mathit{f}(\mathit{z}){|}^2}:\mathit{z}\in {\mathit{f}}^{-1}\{{\mathit{a}}_1,\dots ,{\mathit{a}}_5\}\}<\infty .$$

We give a lower bound for the Hausdorff dimension of the support of the Green current associated with some meromorphic maps.

We determine explicit generators for a cohomology group constructed from a solution of a Fuchsian linear differential equation and describe its relation with cohomology groups with coefficients in a local system. In the parametrized case, this yields into an algorithm which computes new Fuchsian differential equations from those depending on multi-parameters. This generalizes the classical convolution of solutions of Fuchsian differential equations.

For a link in a thickened annulus $\mathit{A}\times \mathit{I}$, we define a $\mathbb{Z}\oplus \mathbb{Z}\oplus \mathbb{Z}$ filtration on Sarkar–Seed–Szabó’s perturbation of the geometric spectral sequence. The filtered chain homotopy type is an invariant of the isotopy class of the annular link. From this, we define a two-dimensional family of annular link invariants and study their behavior under cobordisms. In the case of annular links obtained from braid closures, we obtain a necessary condition for braid quasi-positivity and a sufficient condition for right-veeringness, as well as Bennequin-type inequalities.

Let $(\mathit{X},{\mathit{T}}^{1,0}\mathit{X})$ be a $(2\mathit{n}+1+\mathit{d})$-dimensional compact CR manifold with codimension $\mathit{d}+1$, $\mathit{d}\ge 1$, and let G be a d-dimensional compact Lie group with CR action on X and T be a globally defined vector field on X such that $\mathbb{C}\mathit{T}\mathit{X}={\mathit{T}}^{1,0}\mathit{X}\oplus {\mathit{T}}^{0,1}\mathit{X}\oplus \mathbb{C}\mathit{T}\oplus \mathbb{C}\underset{\_}{\mathfrak{g}}$, where $\underset{\_}{\mathfrak{g}}$ is the space of vector fields on X induced by the Lie algebra of G. In this work, we show that if X is strongly pseudoconvex in the direction of T and $\mathit{n}\ge 2$, then there exists a G-equivariant CR embedding of X into ${\mathbb{C}}^{\mathit{N}}$ for some $\mathit{N}\in \mathbb{N}$. We also establish a CR orbifold version of Boutet de Monvel’s embedding theorem.

Let $\mathit{q}\in \mathbb{C}$, let

and let ${\mathit{G}}_{\mathit{q}}<{\mathrm{SL}}_2(\mathbb{C})$ be the group generated by a and ${\mathit{b}}_{\mathit{q}}$. In this paper, we study the problem of determining when the group ${\mathit{G}}_{\mathit{q}}$ is not free for $|\mathit{q}|<4$ rational. We give a robust computational criterion, which allows us to prove that if $\mathit{q}=\mathit{s}/\mathit{r}$ for $|\mathit{s}|\le 27$, then ${\mathit{G}}_{\mathit{q}}$ is non-free with the possible exception of $\mathit{s}=24$. In this latter case, we prove that the set of denominators $\mathit{r}\in \mathbb{N}$ for which ${\mathit{G}}_{24/\mathit{r}}$ is non-free has natural density 1. For a general numerator $\mathit{s}>27$, we prove that the lower density of denominators $\mathit{r}\in \mathbb{N}$ for which ${\mathit{G}}_{\mathit{s}/\mathit{r}}$ is non-free has a lower bound

$$1-(1-\frac{11}{\mathit{s}})\prod _{\mathit{n}=1}^{\infty }(1-\frac{4}{{\mathit{s}}^{2^{\mathit{n}}-1}}).$$

Finally, we show that for a fixed s, there are arbitrarily long sequences of consecutive denominators r such that ${\mathit{G}}_{\mathit{s}/\mathit{r}}$ is non-free. The proofs of some of the results are computer assisted, and Mathematica code has been provided together with suitable documentation.

Consider an integer $\mathit{n}\ge 2$, $\mathit{m}\in [\mathit{n},+\infty )$ and put $\bar{\mathit{\alpha }}:=\frac{\mathit{m}-1}{\mathit{n}-1}$. Moreover:

∙ Let ${\mathit{\gamma }}_{\mathit{x}}:(0,+\infty )\to {\mathbb{R}}^{\mathit{n}}$ be the line through $\mathit{x}=({\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{n}})\in {\mathbb{R}}^{\mathit{n}}$ defined as ${\mathit{\gamma }}_{\mathit{x}}(\mathit{t}):=(\mathit{t}{\mathit{x}}_1,{\mathit{t}}^{\bar{\mathit{\alpha }}}{\mathit{x}}_2,\dots ,{\mathit{t}}^{\bar{\mathit{\alpha }}}{\mathit{x}}_{\mathit{n}})$;

∙ If T is any motion of ${\mathbb{R}}^{\mathit{n}}$, then let ${\mathit{\gamma }}_{\mathit{x}}^{(\mathit{T})}$ be the line through $\mathit{x}=({\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{n}})\in {\mathbb{R}}^{\mathit{n}}$ defined as ${\mathit{\gamma }}_{\mathit{x}}^{(\mathit{T})}:=\mathit{T}\circ {\mathit{\gamma }}_{{\mathit{T}}^{-1}(\mathit{x})}$;

∙ Let ${\mathcal{H}}^1$ denote the one-dimensional Hausdorff measure in ${\mathbb{R}}^{\mathit{n}}$.

The main goal of this paper is to prove the following property: If ${\mathit{x}}_0$ is an m-density point of a Lebesgue measurable set E and T is an arbitrary motion of ${\mathbb{R}}^{\mathit{n}}$ mapping the origin to ${\mathit{x}}_0$, then we have

$$\underset{\mathit{t}\to 0+}{lim\hspace{0.17em}sup}\frac{{\mathcal{H}}^1(\mathit{E}\cap {\mathit{\gamma }}_{\mathit{x}}^{(\mathit{T})}((0,\mathit{t}]))}{{\mathcal{H}}^1({\mathit{\gamma }}_{\mathit{x}}^{(\mathit{T})}((0,\mathit{t}]))}=1$$

for almost every $\mathit{x}\in \mathit{T}(\{1\}\times {\mathbb{R}}^{\mathit{n}-1})$. An application of this result to locally finite perimeter sets is provided.

We prove local uniformization of Abhyankar valuations of an algebraic function field K over a ground field k. Our result generalizes the proof of this result, with the additional assumption that the residue field of the valuation ring is separable over k, by Hagen Knaf and Franz-Viktor Kuhlmann. The proof in this paper uses different methods, being inspired by the approach of Zariski and Abhyankar.