Equivariant Almost Arborescent Representations Of Open Simply Connected 3 Manifolds A Finiteness Result

When one extends the (almost) collapsible pseudo-spine representation theorem for homotopy $3$-spheres [Po3] to open simply connected $3$-manifolds $V^3$, new phenomena appear: at the source of the representation, the set of double points is, generally speaking, no longer closed. We show that at the cost of replacing $V^3$ by $V_h^3 = \{V^3$ with very many holes $\}$, we can always find representations $X^2 \stackrel {f} {\rightarrow} V^3$ with $X^2$ locally finite and almost-arborescent, with $\Psi (f)=\Phi (f)$, with the open regular neighbourhood (the only one which is well-defined here) Nbd$(fX^2)=V^3_h$ and such that on any precompact tight transversal to the set of double lines, we have only finitely many limit points (of the set of double points).Moreover, if $V^3$ is the universal covering space of a closed $3$-manifold, $V^3=\widetilde M^3$, then we can find an $X^2$ with a free $\pi_1M^3$ action and having the equivariance property $f(gx)=gf(x)$, $g\in \pi_1M^3$. Having simultaneously all these properties for $X^2\stackrel{f} {\rightarrow} \widetilde M^3$ is one of the steps in the first author's program for proving that $\pi_1^\infty \widetilde M^3=[UNK]0$, [Po11, Po12]. Achieving equivariance is far from being straightforward, since $X^2$ is gotten starting from a tree of fundamental domains on which $\pi_1M^3$ cannot, generally speaking, act freely. So, in this paper we have both a representation theorem for general ($\pi_1=0$) $V^3$'s and a harder equivariant representation theorem for $\widetilde M^3$ (with $gfX^2=fX^2, \, g\in\pi_1M^3$), the proof of which is not a specialization of the first, 'easier' result.But, finiteness is achieved in both contexts. In a certain sense, this finiteness is a best possible result, since if the set of limit points in question is $\emptyset$ (i.e. if the set of double points is closed), then $\pi_1^\infty V_h^3$ (which is always equal to $\pi_1^\infty V^3$) is zero. In [PoTa2] it was also shown that when we insist on representing $V^3$ itself, rather than $V_h^3$, and if $V^3$ is wild ($\pi_1^\infty\not =0$), then the transversal structure of the set of double lines can exhibit chaotic dynamical behavior. Our finiteness theorem avoids chaos at the cost of a lot of redundancy (the same double point $(x, y)$ can be reached in many distinct ways starting from the singularities).

Shows that at the cost of replacing $V DEGREES3$ by $V_h DEGREES3 = \{V DEGREES3$ with very many holes $\}$, we can always find representations $X DEGREES2 \stackrel {f} {\rightarrow} V DEGREES3$ with $X DEGREES2$ locally finite and almost-arborescent, with $\Psi (f)=\Phi (f)$, and with the ope

This memoir completes the series of papers beginning with [KL1,KL2], showing that, for a commutative noetherian ring $\Lambda$, either the category of $\Lambda$-modules of finite length has wild representation type or else we can describe the category of finitely generated $\Lambda$-modules, including their direct-sum relations and local-global relations. (There is a possible exception to our results, involving characteristic 2.)

On the assumption that the $\Gamma$-orbits all have dimension equal to that of $\Gamma$, this title shows that there is a naturally defined $F$- and $\Gamma$-invariant measure $\nu$ of maximal entropy on $\Lambda$ (it is not assumed that the action of $\Gamma$ is free).

We develop the basic theory of root systems $R$ in a real vector space $X$ which are defined in analogy to the usual finite root systems, except that finiteness is replaced by local finiteness: the intersection of $R$ with every finite-dimensional subspace of $X$ is finite. The main topics are Weyl groups, parabolic subsets and positive systems, weights, and gradings.

Studies the evolution of the large finite spatial systems in size-dependent time scales and compare them with the behavior of the infinite systems, which amounts to establishing the so-called finite system scheme. This title introduces the concept of a continuum limit in the hierarchical mean field limit.

Aims to introduce the reader to various forms of the maximum principle, starting from its classical formulation up to generalizations of the Omori-Yau maximum principle at infinity obtained by the authors.

Ahlfors conjectured in 1964 that the limit set of every finitely generated Kleinian group either has Lebesgue measure $0$ or is the entire $S^2$. This title intends to prove that this conjecture is true for purely loxodromic Kleinian groups which are algebraic limits of geometrically finite groups.

Intends to complete the determination of the maximal subgroups of positive dimension in simple algebraic groups of exceptional type over algebraically closed fields. This title follows work of Dynkin, who solved the problem in characteristic zero, and Seitz who did likewise over fields whose characteristic is not too small.

Let $\cal{R}$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper the authors establish a sharp threshold for random graphs with this property. Let $G(n, p)$ be the random graph on $n$ vertices with edge probability $p$. The authors prove that there exists a function $\widehat c=\widehat c(n)=\Theta(1)$ such that for any $\varepsilon > 0$, as $n$ tends to infinity, $Pr\left[G(n, (1-\varepsilon)\widehat c/\sqrt{n}) \in \cal{R} \right] \rightarrow 0$ and $Pr \left[ G(n, (1]\varepsilon)\widehat c/\sqrt{n}) \in \cal{R}\ \right] \rightarrow 1.$. A crucial tool that is used in the proof and is of independent interest is a generalization of Szemeredi's Regularity Lemma to a certain hypergraph setti