Let
Let
we answer to a question in Almgren’s
we generalize Almgren’s connectivity theorem showing that the support of
we conclude a structural result on
 Award ID(s):
 1854147
 NSFPAR ID:
 10532757
 Publisher / Repository:
 American Mathematical Society
 Date Published:
 Journal Name:
 Memoirs of the American Mathematical Society
 Volume:
 291
 Issue:
 1446
 ISSN:
 00659266
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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$\Gamma$ be a countable abelian group. An (abstract)$\Gamma$ system$\mathrm {X}$  that is, an (abstract) probability space equipped with an (abstract) probabilitypreserving action of$\Gamma$  is said to be aConze–Lesigne system if it is equal to its second Host–Kra–Ziegler factor$\mathrm {Z}^2(\mathrm {X})$ . The main result of this paper is a structural description of such Conze–Lesigne systems for arbitrary countable abelian$\Gamma$ , namely that they are the inverse limit of translational systems$G_n/\Lambda _n$ arising from locally compact nilpotent groups$G_n$ of nilpotency class$2$ , quotiented by a lattice$\Lambda _n$ . Results of this type were previously known when$\Gamma$ was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers$U^3(G)$ norm for arbitrary finite abelian groups$G$ . 
For each odd integer
$n \geq 3$ , we construct a rank3 graph$\Lambda _n$ with involution$\gamma _n$ whose real$C^*$ algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n)$ is stably isomorphic to the exotic Cuntz algebra$\mathcal E_n$ . This construction is optimal, as we prove that a rank2 graph with involution$(\Lambda ,\gamma )$ can never satisfy$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )\sim _{ME} \mathcal E_n$ , and Boersema reached the same conclusion for rank1 graphs (directed graphs) in [Münster J. Math.10 (2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank1 graph with involution$(\Lambda , \gamma )$ whose real$C^*$ algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )$ is stably isomorphic to the suspension$S \mathbb {R}$ . In the Appendix, we show that the$i$ fold suspension$S^i \mathbb {R}$ is stably isomorphic to a graph algebra iff$2 \leq i \leq 1$ . 
Let
$(G/\Gamma ,R_a)$ be an ergodic$k$ step nilsystem for$k\geq 2$ . We adapt an argument of Parry [Topology 9 (1970), pp. 217–224] to show that$L^2(G/\Gamma )$ decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite multiplicity. In particular, we generalize a result previously established by Host–Kra–Maass [J. Anal. Math.124 (2014), pp. 261–295] for$2$ step nilsystems and a result by Stepin [Uspehi Mat. Nauk24 (1969), pp. 241–242] for nilsystems$G/\Gamma$ with connected, simply connected$G$ . 
We show that for primes
$N, p \geq 5$ with$N \equiv 1 \bmod p$ , the class number of$\mathbb {Q}(N^{1/p})$ is divisible by$p$ . Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when$N \equiv 1 \bmod p$ , there is always a cusp form of weight$2$ and level$\Gamma _0(N^2)$ whose$\ell$ th Fourier coefficient is congruent to$\ell + 1$ modulo a prime above$p$ , for all primes$\ell$ . We use the Galois representation of such a cusp form to explicitly construct an unramified degree$p$ extension of$\mathbb {Q}(N^{1/p})$ . 
In this paper we consider which families of finite simple groups
$G$ have the property that for each$\epsilon > 0$ there exists$N > 0$ such that, if$G \ge N$ and$S, T$ are normal subsets of$G$ with at least$\epsilon G$ elements each, then every nontrivial element of$G$ is the product of an element of$S$ and an element of$T$ .We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form
$\mathrm {PSL}_n(q)$ where$q$ is fixed and$n\to \infty$ . However, in the case$S=T$ and$G$ alternating this holds with an explicit bound on$N$ in terms of$\epsilon$ .Related problems and applications are also discussed. In particular we show that, if
$w_1, w_2$ are nontrivial words,$G$ is a finite simple group of Lie type of bounded rank, and for$g \in G$ ,$P_{w_1(G),w_2(G)}(g)$ denotes the probability that$g_1g_2 = g$ where$g_i \in w_i(G)$ are chosen uniformly and independently, then, as$G \to \infty$ , the distribution$P_{w_1(G),w_2(G)}$ tends to the uniform distribution on$G$ with respect to the$L^{\infty }$ norm.