A Brouwer fixed point theorem for graph endomorphisms

We prove a Lefschetz formula for general simple graphs which equates the Lefschetz number L(T) of an endomorphism T with the sum of the degrees i(x) of simplices in G which are fixed by T. The degree i(x) of x with respect to T is defined as a graded sign of the permutation T induces on the simplex x multiplied by -1 if the dimension of x is odd. The Lefschetz number is defined as in the continuum as the super trace of T induced on cohomology. In the special case where T is the identity, the formula becomes the Euler-Poincare formula equating combinatorial and cohomological Euler characteristic. The theorem assures in general that if L(T) is nonzero, then T has a fixed clique. A special case is a discrete Brouwer fixed point theorem for graphs: if T is a graph endomorphism of a connected graph G, which is star-shaped in the sense that only the zeroth cohomology group is nontrivial, like for connected trees or triangularizations of star shaped Euclidean domains, then there is clique x which is fixed by T. Unlike in the continuum, the fixed point theorem proven here looks for fixed cliques, complete subgraphs which play now the role of "points" in the graph. Fixed points can so be vertices, edges, fixed triangles etc. If A denotes the automorphism group of a graph, we also look at the average Lefschetz number L(G) which is the average of L(T) over A. We prove that this is the Euler characteristic of the graph G/A and especially an integer. We also show that as a consequence of the Lefschetz formula, the zeta function zeta(T,z) is a product of two dynamical zeta functions and therefore has an analytic continuation as a rational function which is explicitly given by a product formula involving only the dimension and the signature of prime orbits of simplices in G.Comment: 24 pages, 6 figure

Residual properties of 1-relator groups

This is a survey of two papers joint with A. Borisov and a paper joint with I. Spakulova. It is based on my lectures at the conference "Groups St. Andrews 2009", Bath (August 2009). We prove that almost all 1-related groups with at least 3 generators are virtually residually (finite p-)groups for almost all primes p, and coherent. The proof involves methods from combinatorial group theory (the congruence extension property of certain subgroups of free groups) algebraic geometry (dynamics of polynomial maps over finite and p-adic fields) and probability theory (convex hulls of Brownian bridges).Comment: 19 pages

Galois symmetries of fundamental groupoids and noncommutative geometry

We define motivic iterated integrals on the affine line, and give a simple proof of the formula for the coproduct in the Hopf algebra of they make. We show that it encodes the group law in the automorphism group of certain non-commutative variety. We relate the coproduct with the coproduct in the Hopf algebra of decorated rooted planar trivalent trees - a planar decorated version of the Hopf algebra defined by Connes and Kreimer. As an application we derive explicit formulas for the coproduct in the motivic multiple polylogarithm Hopf algebra. We give a criteria for a motivic iterated integral to be unramified at a prime ideal, and use it to estimate from above the space spanned by the values of iterated integrals. In chapter 7 we discuss some general principles relating Feynman integrals and mixed motives.Comment: 51 pages, The final version to appear in Duke Math.