Related papers: The limit cone and bounds on the growth indicator …

Let $G=NH$ be a Lie group where $N,H$ are closed connected subgroups of $G,$ and $N$ is an exponential solvable Lie group which is normal in $G.$ Suppose furthermore that $N$ admits a unitary character $\chi_{\lambda}$ corresponding to a…

Let $X$ be a Banach space, let $(\Omega,\mu)$ be a $\sigma$-finite measure space and let $A,B\colon\Omega\to B(X)$ be strongly measurable $\gamma$-bounded functions. We show that for all $x\in X$ and all $x^*\in X^*$, there exist a Hilbert…

Given a lattice \Gamma in a locally compact group G and a closed subgroup H of G, one has a natural action of \Gamma on the homogeneous space V=H\G. For an increasing family of finite subsets {\Gamma_T: T>0}, a dense orbit v\Gamma, v\in V,…

The $2 q$-th pseudomoment $\Psi_{2q,\alpha}(x)$ of the $\alpha$-th power of the Riemann zeta function is defined to be the $2 q$-th moment of the partial sum up to $x$ of $\zeta^\alpha$ on the critical line. Using probabilistic methods of…

We provide a growth bound for the operator norm of $C_0$-semigroups on Hilbert spaces under a corresponding growth bound on the resolvent of the semigroup generator. For some super-linear resolvent growths, our estimate is sharper than the…

We show that at least 1/3 of positive real numbers are in the set of limit points of normalized prime gaps. More precisely, if $p_n$ denotes the $n$th prime and $\mathbb{L}$ is the set of limit points of the sequence $\{(p_{n+1}-p_n)/\log…

The aim of this short note is to revisit some old results about groups of intermediate growth and groups of the lamplighter type and to show that the Lamplighter group $L=\mathbb{Z}_2\wr \mathbb{Z}$ is a condensation group and has a minimal…

We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing…

Let $(G,K)$ be a Gelfand pair, with $G$ a Lie group of polynomial growth, and let $\Sigma\subset{\mathbb R}^\ell$ be a homeomorphic image of the Gelfand spectrum, obtained by choosing a generating system $D_1,\dots,D_\ell$ of $G$-invariant…

We define the class of groups of bounded type from tile inflations. These tile inflations also determine some automata describing the groups. In the case when the automata are stationary, we show that if the set of incompressible elements…

We announce an explicit description of the strictly semistable boundary of the GIT moduli space of quintic threefolds. For the natural action of \(\mathrm{SL}(5)\) on \(\mathbb P(\mathrm{Sym}^5\mathbb C^5)\), we classify the 38 boundary…

In this note, we show how to adapt Tao's slice rank method to extend the Ellenberg--Gijswijt theorem on cap sets to the problem of forbidding arithmetic progressions with restricted differences. In particular, we show that if $q$ is an odd…

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ \Pi $-property in $ G $ if if there exists a chief series $ \varGamma_{G}: 1 =G_{0} < G_{1} < \cdot\cdot\cdot < G_{n}= G $ of $ G $ such that for…

Let $G$ be a group which is the semidirect product of a normal subgroup $N$ and some subgroup $T$. Let $I^n(G)$, $n\ge 1$, denote the powers of the augmentation ideal $I(G)$ of the group ring $\Z(G)$. Using homological methods the groups…

Let $G$ be a compact Lie group, $N\geq 1$ and $L>0$. The random geometric graph on $G$ is the random graph $\Gamma(N,L)$ whose vertices are $N$ random points $g_1,\ldots,g_N$ chosen under the Haar measure of $G$, and whose edges are the…

Let N be an irreducible, compact 3-manifold with empty or toroidal boundary which is not a closed graph manifold. Using recent work of Agol, Kahn-Markovic and Przytycki-Wise we will show that pi_1(N) admits a cofinal filtration with `fast'…

Let $S$ be subsemigroup with nonempty interior of a complex simple Lie group $G$. It is proved that $S=G$ if $S$ contains a subgroup $G(\alpha) \approx \mathrm{Sl}(2,\mathbb{C}) $ generated by the $\exp \mathfrak{g}_{\pm \alpha}$, where…

Given an abstract group $G$, we study the function $ab_n(G) := \sup_{|G:H| \leq n} |H/[H,H]|$. If $G$ has no abelian composition factors, then $ab_n(G)$ is bounded by a polynomial: as a consequence, we find a sharp upper bound for the…

We show that there exists a finitely generated group of growth ~f for all functions f:\mathbb{R}\rightarrow\mathbb{R} satisfying f(2R) \leq f(R)^{2} \leq f(\eta R) for all R large enough and \eta\approx2.4675 the positive root of…

For a finitely generated lawless group $\Gamma$ and $n \in \mathbb{N}$, let $\mathcal{A}_{\Gamma} (n)$ be the minimal positive integer $M_n$ such that for all nontrivial reduced words $w$ of length at most $n$ in the free group of fixed…