Let $\Gamma < \operatorname{PSL}_2(\mathbb{R})= \text{Isom}^+(\mathbb{H^2})$ be a discrete subgroup. Suppose $\Gamma$ acts ergodically on the boundary of the hyperbolic plane $\partial{\mathbb{H}^2}$. Does it follow that any finite index subgroup $\Gamma_0$ of $\Gamma$ also acts ergodically on $\...

Let $A_n$ be a random $n \times n$ matrix with entries in $\{-1, +1\}$. As usual, "random" here means with respect to the uniform measure over such matrices. The strong version of the singularity conjecture is that, as $n \to \infty$, $$ \mathbb{P}\{ \text{det}(A_n) = 0\} = (1 +o(1)) \, n^2 2^{-(...

Problem Let $\beta$ be a probability measure on $\mathbb{R}$, and define $$ K = \left \{z \in \mathbb{C}: g\left(z\right)=\int_{-\infty}^{\infty}\exp\left(z x\right)\beta ( dx ) \text{ is well-defined} \right\} $$ How do we do know if $g$ has any zeros in $K$? What are the conditions to ensure ...

Looking for a reference for the following easy-to-prove fact: Say $T$ and $S$ are monads on $\text{Set}$ admitting a monoid homomorphism $\phi : S \to T$ (i.e., a morphism in $\text{Mon}([\text{Set},\text{Set}])$. Then any $T$-algebra is also an $S$-algebra. I imagine this might be stated in the ...

Let $\mathcal{F}\subseteq2^{[n]},\emptyset\in\mathcal{F}$ be an union-closed family of sets. For $S\in\mathcal{F}$, let $w(S)$ be the number of subsets of $S$ in $\mathcal{F}$. Does there always exist real numbers $a_1,a_2,...,a_n\geq1$ such that $\prod_{i=1}^na_i=|\mathcal{F}|$ and for every $S\...

In a paper Small doubling in ordered groups: generators and structure it is proven in Lemma 4 page no. 598 that: Let $G$ be an ordered group. Let $S$ be a finite subset of $G$ with at least two elements. Let either $ m=$ max $S$ or $m=$ min $S$ and $T= S\setminus\{m\}.$ Then either $\langle S\ra...

Let $(X,\mathcal{T})$ be a connected topological space. Let $\mathcal{T}'$ be the topology on $X$ that is generated by the collection of connected open sets in $(X,\mathcal{T})$. That is, the connected open sets in $(X,\mathcal{T})$ form a subbasis for $\mathcal{T}'$. Is it necessarily true that ...