Related papers: Non-rectifiable Delone sets in SOL and other solva…

Certain semigroups are known to admit a `strong semilattice decomposition' into simpler pieces. We introduce a class of Banach algebras that generalise the $\ell^1$-convolution algebras of such semigroups, and obtain a disintegration…

Let $\Gamma$ be a lattice in $\mathrm{SO}_0(n, 1)$. We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least $2$, then $\Gamma$ is arithmetic. This answers a…

In this paper we explore the interplay between aspects of the geometry and algebra of three families of groups of the form B semidirect the integers Z, namely Lamplighter groups, solvable Baumslag-Solitar groups and lattices in SOL. In…

Motivated by a question of M. Kapovich, we show that the $\mathbb{Z}^2$ subgroups of $\mathsf{SL}_3(\mathbb{R})$ that are regular in the language of Kapovich--Leeb--Porti, or divergent in the sense of Guichard--Wienhard, are precisely the…

Let $\Gamma$ be a group acting on with finite stabilizers and finite fundamental domain on a building of type $\tilde A_2$. We prove that any non-trivial normal subgroup of $\Gamma$ is of finite index in $\Gamma$.

We show a rigidity result for subfactors that are normalized by a representation of a lattice $\Gamma$ in a higher rank simple Lie group with trivial center into a finite factor. This implies that every subfactor of $L\Gamma$ which is…

Let Gamma be an S-arithmetic subgroup of a solvable algebraic group G over an algebraic number field F, such that the finite set S contains at least one place that is nonarchimedean. We construct a certain group H, such that if L is any…

For every $k\geqslant 3$, we exhibit a simply connected $k$-nilpotent Lie group $N_k$ whose Dehn function behaves like $n^k$, while the Dehn function of its associated Carnot graded group $\mathsf{gr}(N_k)$ behaves like $n^{k+1}$. This…

We consider nonhomogeneous fractional $p$-Laplace equations defined on a bounded nonsmooth domain which goes beyond the Lipschitz category. Under a sufficient flatness assumption on the domain in the sense of Reifenberg, we establish…

We show that an arithmetic lattice $\Gamma$ in a semi-simple Lie group $G$ contains a torsion-free subgroup of index $\delta(v)$ where $v = \mu (G/\Gamma)$ is the co-volume of the lattice. We prove that $\delta$ is polynomial in general and…

We describe a family of finitely presented groups which are quasi-isometric but not bilipschitz equivalent. The first such examples were described by the first author and are the lamplighter groups $F \wr \mathbb{Z}$ where $F$ is a finite…

Let $G$ be a real Lie group and $\Gamma < G$ be a discrete subgroup of $G$. Is $\Gamma$ residually finite? This paper describes known positive and negative results then poses some questions whose answers will lead to a fairly complete…

The Chermak-Delgado lattice of a finite group $G$ is a self-dual sublattice of the subgroup lattice of $G$. In this paper, we focus on finite groups whose Chermak-Delgado lattice is a subgroup lattice of an elementary abelian $p$-group. We…

The Drinfeld double D of the bosonization of a finite-dimensional Nichols algebra B(V) over a finite non-abelian group G is called a quantum group at a non-abelian group. We introduce Verma modules over such a quantum group D and prove that…

We continue and complete our previous paper `Lifts of projective congruence groups' [2] concerning the question of whether there exist noncongruence subgroups of $\SL_2(\Z)$ that are projectively equivalent to one of the groups…

This paper gives a new explicit construction of the $\mathbb{Q}$-algebraic hull for virtually solvable groups $\Gamma$ of finite abelian ranks, taking into account the spectrum $S$ of the group $\Gamma$. As an application, we make a…

In this paper we prove that, for any integer $d>0$, every linearly repetitive Delone set in the Euclidean $d$-space $\RR^d$ is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice $\ZZ^d$. In the particular case when the…

Let $\Gamma$ be a finitely generated group which is hyperbolic relative to a finite family $\{H_1,...,H_n\}$ of subgroups. We prove that $\Gamma$ is uniformly embeddable in a Hilbert space if and only if each subgroup $H_i$ is uniformly…

We prove that the noncrossing partition lattices associated with the complex reflection groups $G(d,d,n)$ for $d,n\geq 2$ admit symmetric decompositions into Boolean subposets. As a result, these lattices have the strong Sperner property…

In this communication, the co-maximal subgroup graph $\Gamma(G)$ of a finite group $G$ is examined when $G$ is a finite nilpotent group, finite abelian group, dihedral group $D_n$, dicyclic group $Q_{2^n}$, and $p$-group. We derive the…