By Robert Bieri

ISBN-10: 0821831844

ISBN-13: 9780821831847

Generalizing the Bieri-Neumann-Strebel-Renz Invariants, this Memoir offers the principles of a concept of (not unavoidably discrete) activities $\rho$ of a (suitable) crew $G$ by way of isometries on a formal CAT(0) house $M$. The passage from teams $G$ to staff activities $\rho$ implies the creation of 'Sigma invariants' $\Sigma^k(\rho)$ to switch the former $\Sigma^k(G)$ brought by means of these authors. Their conception is now obvious as a unique case of what's studied right here in order that readers looking an in depth therapy in their conception will locate it incorporated the following as a unique case. We outline and research 'controlled $k$-connectedness $(CC^k)$' of $\rho$, either over $M$ and over finish issues $e$ within the 'boundary at infinity' $\partial M$; $\Sigma^k(\rho)$ is by way of definition the set of all $e$ over which the motion is $(k-1)$-connected. A relevant theorem, the Boundary Criterion, says that $\Sigma^k(\rho) = \partial M$ if and provided that $\rho$ is $CC^{k-1}$ over $M$.An Openness Theorem says that $CC^k$ over $M$ is an open situation at the house of isometric activities $\rho$ of $G$ on $M$. one other Openness Theorem says that $\Sigma^k(\rho)$ is an open subset of $\partial M$ with admire to the titties metric topology. whilst $\rho(G)$ is a discrete team of isometries the valuables $CC^{k-1}$ is akin to ker$(\rho)$ having the topological finiteness estate style '$F_k$'. extra ordinarily, if the orbits of the motion are discrete, $CC^{k-1}$ is reminiscent of the point-stabilizers having kind $F_k$. particularly, for $k=2$ we're characterizing finite presentability of kernels and stabilizers. Examples mentioned contain: in the community inflexible activities, translation activities on vector areas (especially these by means of metabelian groups), activities on timber (including these of $S$-arithmetic teams on Bruhat-Tits trees), and $SL_2$ activities at the hyperbolic aircraft.

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Be independent Bernoulli random variables, with P(Xn = −1) = P(Xn = 1) = 12 for every n, and set Sn √= ni=1 Xi . The law of the iterated logarithm states that lim supn→∞ Sn / 2n log log n = 1 almost surely. Putting it another way, for t ∈ [0, 1], let t = 0. 1 (t) 2 (t) . . be its dyadic expansion, or equivalently, set n (t) = 0 or 1 according as the integer part of 2n t is even or odd. ) Set fn (t) = nk=1 k (t) − n2 . Then the law of iterated logarithm states that fn (t) =1 lim sup n 1/2 n→∞ ( 2 log log n) for almost every t ∈ (0, 1].

Number of edges) of a graph G. Given graphs G and H, the expression H ⊂ G means that H is a subgraph of G. Let F be a ﬁxed graph, usually called the forbidden graph. Set ex(n; F ) = max{e(G) : |G| = n and F ⊂ G}. Paul Erd˝ os: Life and Work 27 and EX(n; F ) = {G : |G| = n, e(G) = ex(n; F ), and F ⊂ G}. We call ex (n; F ) the extremal function, and EX (n; F ) the set of extremal graphs for the forbidden graph F . Then the basic problem of extremal graph theory is to determine, or at least estimate, ex(n; F ) for a given graph F and, at best, to determine EX(n; F ).

Note that if all we care about is whether two r-sets belong to the same class or not, then for every ordered set X with more than r elements, X (r) has precisely 2r distinct canonical partitions, one for each subset I of [r]. If X is inﬁnite then there is only one canonical partition with ﬁnitely many classes: this is the canonical partition belonging to I = ∅, in which all r-sets belong to the same class. Paul Erd˝ os: Life and Work 33 The Erd˝ os-Rado canonical Ramsey theorem claims that if we partition N(r) into any number of classes then there is always an inﬁnite sequence of integers x1 < x2 < .