We study II${}_1$ factors $M$ and $N$ associated with good generalized Bernoulli actions of groups having an infinite almost normal subgroup with the relative property (T). We prove the following rigidity result : every finite index $M$-$N$-bimodule (in particular, every isomorphism between $M$ and $N$) is described by a commensurability of the groups involved and a commensurability of their actions. The fusion algebra of finite index $M$-$M$-bimodules is identified with an extended Hecke fusion algebra,...

For any $2\le n\le \infty$, we construct a concrete 1-parameter family of non-orbit equivalent actions of the free group ${𝔽}_n$. These actions arise as diagonal products between a generalized Bernoulli action and the action ${𝔽}_n\curvearrowright ({𝕋}^2,{\lambda }^2)$, where ${𝔽}_n$ is seen as a subgroup of ${\mathrm{SL}}_2\left(\mathbb{Z}\right)$.

Ozawa showed in [21] that for any i.c.c. hyperbolic group, the associated group factor $L\Gamma$ is solid. Developing a new approach that combines some methods of Peterson [29], Ozawa and Popa [27, 28], and Ozawa [25], we strengthen this result by showing that $L\Gamma$ is strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana [12], we show that profinite actions of lattices in $\mathrm{Sp}(n,1)$, $n\ge 2$, are virtually $W^{*}$-superrigid.

Let $F=(f_1,...,f_q)$ be a polynomial dominating map from ${ℂ}^n$ to ${ℂ}^q$. We study the quotient ${𝒯}^1\left(F\right)$ of polynomial 1-forms that are exact along the generic fibres of $F$, by 1-forms of type $\mathrm{d}R+\sum a_i\mathrm{d}{f}_i$, where $R,a_1,...,a_q$ are polynomials. We prove that ${𝒯}^1\left(F\right)$ is always a torsion $ℂ[t_1,...,t_q]$-module. Then we determine under which conditions on $F$ we have ${𝒯}^1\left(F\right)=0$. As an application, we study the behaviour of a class of algebraic $({ℂ}^p,+)$-actions on ${ℂ}^n$, and determine in particular when these actions are trivial.

We describe the finite group actions, up to equivalence, which can act on the orbifold $\Sigma (2,2,2)$, and their quotient types. This is then used to consider actions on prism manifolds $M(b,d)$ which preserve a longitudinal fibering, but do not leave any Heegaard Klein bottle invariant. If $\varphi :G\to \text{Homeo}\left(M\right(b,d\left)\right)$ is such an action, we show that $M(b,d)=M(b,2)$ and $M(b,2)/\varphi$ fibers over a certain collection of 2-orbifolds with positive Euler characteristic which are covered by $\Sigma (2,2,2)$. For the standard actions, we compute the fundamental group of $M(b,2)/\varphi$ and...

Let $k$ be an algebraically closed field of characteristic $p\>0$. We study obstructions to lifting to characteristic $0$ the faithful continuous action $\phi$ of a finite group $G$ on $k\left[\right[t\left]\right]$. To each such $\phi$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $\phi$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic $0$ in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H\subset G$. We determine for which $G$ the KGB...

We consider both standard and twisted actions of a (real) Coxeter group $G$ on the complement ${ℳ}_G$ to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in $G$ and give explicit formulae which describe both actions on the total cohomology $H^{*}({ℳ}_G,𝒞)$ in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group $S_n$, the...

Let Fⁿ be a connected, smooth and closed n-dimensional manifold satisfying the following property: if $N^m$ is any smooth and closed m-dimensional manifold with m > n and $T:N^m\to N^m$ is a smooth involution whose fixed point set is Fⁿ, then m = 2n. We describe the equivariant cobordism classification of smooth actions $(M^m;\Phi )$ of the group $G=Z{₂}^k$ on closed smooth m-dimensional manifolds $M^m$ for which the fixed point set of the action is a submanifold Fⁿ with the above property. This generalizes a result of F....

It is known that $ℬ\left({\ell }^p\right)$ is not amenable for p = 1,2,∞, but whether or not $ℬ\left({\ell }^p\right)$ is amenable for p ∈ (1,∞) ∖ 2 is an open problem. We show that, if $ℬ\left({\ell }^p\right)$ is amenable for p ∈ (1,∞), then so are ${\ell }^{\infty }\left(ℬ\left({\ell }^p\right)\right)$ and ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$. Moreover, if ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$ is amenable so is ${\ell }^{\infty }(,\left(E\right))$ for any index set and for any infinite-dimensional ${ℒ}^p$-space E; in particular, if ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$ is amenable for p ∈ (1,∞), then so is ${\ell }^{\infty }\left(\left({\ell }^p\oplus \ell ²\right)\right)$. We show that ${\ell }^{\infty }\left(\left({\ell }^p\oplus \ell ²\right)\right)$ is not amenable for p = 1,∞, but also that our methods fail us if p ∈ (1,∞). Finally, for p ∈ (1,2) and a free ultrafilter over...

Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $aS$-group. We study some properties of $aS$-groups. For instance, it is shown that a nilpotent group $G$ is an $aS$-group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $aS$-group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group...

Let $k$ be a field of characteristic $p\>0$. Let $D_m$ be a ${BT}_m$ over $k$ (i.e., an $m$-truncated Barsotti–Tate group over $k$). Let $S$ be a $k$-scheme and let $X$ be a ${BT}_m$ over $S$. Let $S_{D_m}\left(X\right)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}\left(X\right)$ is pure in $S$, i.e. the immersion $S_{D_m}\left(X\right)\hookrightarrow S$ is affine. For $p\in \{2,3\}$, we prove purity if $D_m$ satisfies a certain technical property depending only on its $p$-torsion $D_m\left[p\right]$. For $p\ge 5$, we apply the developed techniques to show that...

It is well-known that a probability measure $\mu$ on the circle $𝕋$ satisfies $\parallel {\mu }^n{*f-\int f\mathrm{d}m\parallel }_p\to 0$ for every $f\in L_p$, every (some) $p\in [1,\infty )$, if and only if $|\widehat{\mu }\left(n\right)|lt;1$ for every non-zero $n\in \mathbb{Z}$ ($\mu$ is strictly aperiodic). In this paper we study the a.e. convergence of ${\mu }^n*f$ for every $f\in L_p$ whenever $pgt;1$. We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of $\mu$, for the strong sweeping out property (existence of a Borel set $B$ with $lim\; sup{\mu }^n*1_B=1$ a.e. and $lim\; inf{\mu }^n*1_B=0$ a.e.). The results are extended to general compact Abelian groups...

Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion $X{{(}_k)}^{ét,Br}\subset X{{(}_k)}^{Br₁}$. In the first part, we apply ideas from the proof of $X{{(}_k)}^{ét,Br}=X{{(}_k)}^{{}_k}$ by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if $\subset {\subset }_k$ are such that $\subset Ext\left({,}_k\right)$, then $X{{(}_k)}^{}=X{{(}_k)}^{}$. This allows us to conclude, among other things, that $X{{(}_k)}^{ét,Br}=X{{(}_k)}^{{}_k}$ and $X{{(}_k)}^{Sol,Br₁}=X{{(}_k)}^{So{l}_k}$.

Let $G$ be a group and $m$ be an integer greater than or equal to $2$. $G$ is said to be $m$-permutable if every product of $m$ elements can be reordered at least in one way. We prove that, if $G$ has a centre of finite index $z$, then $G$ is $(1+[z/2])$-permutable. More bounds are given on the least $m$ such that $G$ is $m$-permutable.

In this note we shall prove that for a continuous function $\varphi :\Delta \to {ℝ}^n$, where $\Delta \subset ℝ$,  the paratingent of $\varphi$ at $a\in \Delta$ is a non-empty and compact set in ${ℝ}^n$ if and only if $\varphi$ satisfies Lipschitz condition in a neighbourhood of $a$. Moreover, in this case the paratingent is a connected set.

Let $G$ be a finite group and let $\pi \left(G\right)=\{p_1,p_2,...,p_k\}$ be the set of prime divisors of $\left|G\right|$ for which $p_1<p_2<\cdots <p_k$. The Gruenberg-Kegel graph of $G$, denoted $GK\left(G\right)$, is defined as follows: its vertex set is $\pi \left(G\right)$ and two different vertices $p_i$ and $p_j$ are adjacent by an edge if and only if $G$ contains an element of order $p_i{p}_j$. The degree of a vertex $p_i$ in $\mathrm{GK}\left(G\right)$ is denoted by $d_G\left(p_i\right)$ and the $k$-tuple $D\left(G\right)=(d_G\left(p_1\right),d_G\left(p_2\right),...,d_G\left(p_k\right))$ is said to be the degree pattern of $G$. Moreover, if $\omega \subseteq \pi \left(G\right)$ is the vertex set of a connected component of $GK\left(G\right)$, then the largest $\omega$-number which divides $\left|G\right|$, is...

In this paper, we introduce and study the notion of completely bounded ${\Lambda }_p$ sets (${\Lambda }_p^{cb}$ for short) for compact, non-abelian groups G. We characterize ${\Lambda }_p^{cb}$ sets in terms of completely bounded $L^p\left(G\right)$ multipliers. We prove that when G is an infinite product of special unitary groups of arbitrarily large dimension, there are sets consisting of representations of unbounded degree that are ${\Lambda }_p$ sets for all p < ∞, but are not ${\Lambda }_p^{cb}$ for any p ≥ 4. This is done by showing that the space of completely bounded $L^p\left(G\right)$...

In this note we study sets of normal generators of finitely presented residually $p$-finite groups. We show that if an infinite, finitely presented, residually $p$-finite group $G$ is normally generated by $g_1,\cdots ,g_k$ with order $n_1,\cdots ,n_k\in \{1,2,\cdots \}\cup \left\{\infty \right\}$, then $${\beta }_1^{\left(2\right)}\left(G\right)\le k-1-\sum _{i=1}^k\frac{1}{n_i},$$ where ${\beta }_1^{\left(2\right)}\left(G\right)$ denotes the first ${\ell }^2$-Betti number of $G$. We also show that any $k$-generated group with ${\beta }_1^{\left(2\right)}\left(G\right)\ge k-1-\epsilon$ must have girth greater than or equal $1/\epsilon$.

We study the structure of longest sequences in $\mathbb{Z}{ₙ}^d$ which have no zero-sum subsequence of length n (or less). We prove, among other results, that for $n=2^a$ and d arbitrary, or $n=3^a$ and d = 3, every sequence of c(n,d)(n-1) elements in $\mathbb{Z}{ₙ}^d$ which has no zero-sum subsequence of length n consists of c(n,d) distinct elements each appearing n-1 times, where $c(2^a,d)=2^d$ and $c(3^a,3)=9$.

Let $G$ be a group and $m$ be an integer greater than or equal to $2$. $G$ is said to be $m$-permutable if every product of $m$ elements can be reordered at least in one way. We prove that, if $G$ has a centre of finite index $z$, then $G$ is $(1+[z/2])$-permutable. More bounds are given on the least $m$ such that $G$ is $m$-permutable.