A. Rosenfeld [23] introduced the notion of a digitally continuous function between digital images, and showed that although digital images need not have fixed point properties analogous to those of the Euclidean spaces modeled by the images, there often are approximate fixed point properties of such images. In the current paper, we obtain additional results concerning fixed points and approximate fixed points of digitally continuous functions. Among these are several results concerning the relationship between universal functions and the approximate fixed point property (AFPP).2010 MSC: Primary 55M20; Secondary 55N35.
A discrete group which admits a faithful, finite dimensional, linear representation over a field F of characteristic zero is called linear. This note combines the natural structure of semi-direct products with work of A. Lubotzky [13] on the existence of linear representations to develop a technique to give sufficient conditions to show that a semi-direct product is linear.Let G denote a discrete group which is a semi-direct product given by a split extensionThis note defines an additional type of structure for this semi-direct product called a stable extension below. The main results are as follows:(1) If π and Γ are linear, and the extension is stable, then G is also linear.Restrictions concerning this extension are necessary to guarantee that G is linear as seen from properties of the Formanek-Procesi "poison group" [7]. (2) If the action of Γ on π has a "Galois-like" property that it factors through the automorphisms of certain natural "towers of groups over π" ( to be defined below ), then the associated extension is stable and thus G is linear. (3) The condition of a stable extension also implies that G admits filtration quotients which themselves give a natural structure of Lie algebra and
Let R be a commutative ring that is free of rank k as an abelian group, p a prime, and SLn(R) the special linear group. We show that the Lie algebra associated to the filtration of SLn(R) by p-congruence subgroups is isomorphic to the tensor product sln(R ⊗ Z Z/p) ⊗ Fp tFp[t], the Lie algebra of polynomials with zero constant term and coefficients n × n traceless matrices with entries polynomials in k variables over Fp.We use the Lie algebra structure along with the Lyndon-Hochschild-Serre spectral sequence to compute the d 2 homology differential for certain central extensions involving quotients of p-congruence subgroups. We also use the underlying group structure to obtain several homological results. For example, we compute the first homology group of the level p-congruence subgroup for n ≥ 3. We show that the cohomology groups of the level p r -congruence subgroup are not finitely generated for n = 2 and R = Z[t]. Finally, we show that for n = 2 and R = Z[i] (the Gaussian integers) the second cohomology group of the level p r -congruence subgroup has dimension at least two as an Fp-vector space. We now collect the notation that will be used throughout this paper: R is a commutative ring that is free of rank k as an abelian group; V = {v i } i∈I is a Zbasis for R; G n (R) is an n × n matrix group with coefficients in R; p is a prime; and Γ(G n (R), p r ) is the kernel of the mod-p r reduction map G n (R) − −−− → G n (R ⊗ Z Z/p r ).(2.1)The following is a classical result (see, for example, [7]):
Given a real n × m matrix B, its operator norm can be defined asWe consider a matrix "small" if it has non-negative integer entries and its operator norm is less than 2. These matrices correspond to bipartite graphs with spectral radius less than 2, which can be classified as disjoint unions of Coxeter graphs. This gives a direct route to an ADE-classification result in terms of very basic mathematical objects. Our goal here is to see these results as part of a general program of classification of small objects, relating quadratic forms, reflection groups, root systems, and Lie algebras.2010 AMS Mathematics subject classification. 05C50,15A60,15B36,17B20,17B22,20F55.
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