Fundamental domains for properly discontinuous affine groups

We construct a fundamental region for the action on the \(2d+1\) -dimensional affine space of some free, discrete, properly discontinuous groups of affine transformations preserving a quadratic form of signature \((d+1, d)\) , where \(d\) is any odd positive integer.     

On the properly discontinuous subgroups of affine motions

The fundamental groupΓ of a compact complete affine manifold is represented as an affine crystallographic subgroup of Aff(n). L.S.Auslander conjectured thatΓ is virtually solvable. Our purpose is to find the algebraic condition onΓ which leads affirmative answer to the conjecture.     

Properly discontinuous group actions on affine homogeneous spaces

Let G be a real algebraic group, H ≤ G an algebraic subgroup containing a maximal reductive subgroup of G, and Γ a subgroup of G acting on G/H by left translations. We conjecture that Γ is virtually solvable provided its action on G/H is properly discontinuous and ΓG/H is compact, and we confirm this conjecture when G does not contain simple algebraic subgroups of rank ≥2. If the action of Γ on G/H (which is isomorphic to an affine linear space Aⁿ) is linear, our conjecture coincides with the Auslander conjecture. We prove the Auslander conjecture for n ≤ 5.     

The linear part of an affine group acting properly discontinuously and leaving a quadratic form invariant

In this paper we study the dynamics of properly discontinuous and crystallographic affine groups leaving a quadratic from of signature (p, q) invariant. The main results are: (I) If p − q ≥ 2, then the linear part of the group is not Zariski dense in the corresponding orthogonal group. (II) If q = 2 and the group is crystallographic, then the group is virtually solvable. This proves the Auslander conjecture for this case.     

Complete affine locally flat manifolds with a free fundamental group

Examples of free noncommutative subgroups of the affine group A(3), which act properly discontinuously on ³, are constructed in the paper. These examples refute a conjecture of Milnor to the effect that the fundamental group of any complete affine locally flat manifold contains a sòlvable subgroup of finite index.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 134, pp. 190–205, 1984.     

Expanders obtained from affine transformations

A bipartite graphG=(U, V, E) is an (n, k, δ, α) expander if |U|=|V|=n, |E|≦kn, and for anyX⊆U with |X|≦αn, |Γ _G (X)|≧(1+δ(1−|X|/n)) |X|, whereΓ _G (X) is the set of nodes inV connected to nodes inX with edges inE. We show, using relatively elementary analysis in linear algebra, that the problem of estimating the coefficientδ of a bipartite graph is reduced to that of estimating the second largest eigenvalue of a matrix related to the graph. In particular, we consider the case where the bipartite graphs are defined from affine transformations, and obtain some general results on estimating the eigenvalues of the matrix by using the discrete Fourier transform. These results are then used to estimate the expanding coefficients of bipartite graphs obtained from two-dimensional affine transformations and those obtained from one-dimensional ones.     

Image partition regularity of affine transformations

If u,v∈N, A is a u×v matrix with entries from Q, and , then determines an affine transformation from Qv to Qu by . In 1933 and 1943 Richard Rado determined precisely when such transformations are kernel partition regular over N, Z, or Q, meaning that whenever the nonzero elements of the relevant set are partitioned into finitely many cells, there is some element of the kernel of the transformation with all of its entries in the same cell. In 1993 the first author and Imre Leader determined when such transformations with are image partition regular over N, meaning that whenever N is partitioned into finitely many cells, there is some element of the image of the transformation with all of its entries in the same cell. In this paper we characterize the image partition regularity of such transformations over N, Z, or Q for arbitrary .     

On a free group of transformations defined by an automaton

We prove that three automorphisms of the rooted binary tree defined by a certain 3-state automaton generate a free non-Abelian group of rank 3. Both authors are supported by the NSF grants DMS-0308985 and DMS-0456185. Yaroslav Vorobets is supported by a Clay Research Scholarship.     

A family of affine Weyl group representations

In this paper we explicitly determine the virtual representations of the finite Weyl subgroups of the affine Weyl group on the cohomology of the space of affine flags containing a family of elementsn _t in an affine Lie algebra. We also compute the Euler characteristic of the space of partial flags containingn _t and give a connection with hyperplane arrangements.This paper forms part of my Ph.D. thesis at M.I.T. I was supported by an NSF Graduate Fellowship and NSF grant DMS 9304580.     

De Bruijn digraphs and affine transformations

Let be the additive group of 1×n row vectors over . For an n×n matrix T over  and , the affine transformation F_T,ω of sends x to xT+ω. Let α be the cyclic group generated by a vector . The affine transformation coset pseudo-digraph has the set of cosets of α in as vertices and there are c arcs from x+α to y+α if and only if the number of zx+α such that F_T,ω(z)y+α is c. We prove that the following statements are equivalent: (a)  is isomorphic to the d-nary (n−1)-dimensional De Bruijn digraph; (b) α is a cyclic vector for T; (c)  is primitive. This strengthens a result conjectured by C.M. Fiduccia and E.M. Jacobson [Universal multistage networks via linear permutations, in: Proceedings of the 1991 ACM/IEEE Conference on Supercomputing, ACM Press, New York, 1991, pp. 380–389]. Under the further assumption that T is invertible we show that each component of is a conjunction of a cycle and a De Bruijn digraph, namely a generalized wrapped butterfly. Finally, we discuss the affine TCP digraph representations for a class of digraphs introduced by D. Coudert, A. Ferreira and S. Perennes [Isomorphisms of the De Bruijn digraph and free-space optical networks, Networks 40 (2002) 155–164].     

Ergodic affine transformations are loosely Bernoulli

We prove that an ergodic affine transformation of a compact abelian group is loosely Bernoulli, that is, it can be induced from a Bernoulli shift.     

Algebraic monoids with affine unit group are affine

In this short paper we prove that any irreducible algebraic monoid whose unit group is an affine algebraic group is affine.     

On a functional equation concerning affine transformations

In this paper we prove that, if the functional equation u(Rx + p) = α(R, p) u(x) + β(R, p) has a nonconstant solution u, then α and β have to be of a very special form. Finally, we obtain, under suitable hypotheses, the general solution of this equation.