Various L2-signatures and a topological L2-signature theorem

For a normal covering over a closed oriented topological manifold we give a proof of the L2-signature theorem with twisted coefficients, using Lipschitz structures and the Lipschitz signature operator introduced by Teleman. We also prove that the L-theory isomorphism conjecture as well as the C^*_max-version of the Baum-Connes conjecture imply the L2-signature theorem for a normal covering over a Poincar space, provided that the group of deck transformations is torsion-free. We discuss the various possible definitions of L2-signatures (using the signature operator, using the cap product of differential forms, using a cap product in cellular L2-cohomology,...) in this situation, and prove that they all coincide.Comment: comma in metadata (author field) added

Approximating L^2-signatures by their compact analogues

:Let G be a group together with an descending nested sequence of normal subgroups G=G_0, G_1, G_2 G_3, ... of finite index [G:G_k] such the intersection of the G_k-s is the trivial group. Let (X,Y) be a compact 4n-dimensional Poincare' pair and p: (\bar{X},\bar{Y}) \to (X,Y) be a G-covering, i.e. normal covering with G as deck transformation group. We get associated $G/_k$-coverings (X_k,Y_k) \to (X,Y). We prove that sign^{(2)}(\bar{X},\bar{Y}) = lim_{k\to\infty} \frac{sign(X_k,Y_k)}{[G : G_k]}, where sign or sign^{(2)} is the signature or L^2-signature, respectively, and the convergence of the right side for any such sequence (G_k)_k is part of the statement

Playing Games in the Baire Space

We solve a generalized version of Church's Synthesis Problem where a play is given by a sequence of natural numbers rather than a sequence of bits; so a play is an element of the Baire space rather than of the Cantor space. Two players Input and Output choose natural numbers in alternation to generate a play. We present a natural model of automata ("N-memory automata") equipped with the parity acceptance condition, and we introduce also the corresponding model of "N-memory transducers". We show that solvability of games specified by N-memory automata (i.e., existence of a winning strategy for player Output) is decidable, and that in this case an N-memory transducer can be constructed that implements a winning strategy for player Output.Comment: In Proceedings Cassting'16/SynCoP'16, arXiv:1608.0017

L^2-torsion of hyperbolic manifolds of finite volume

Suppose $\bar{M}$ is a compact connected odd-dimensional manifold with boundary, whose interior $M$ comes with a complete hyperbolic metric of finite volume. We will show that the $L^2$-topological torsion of $\bar{M}$ and the $L^2$-analytic torsion of the Riemannian manifold $M$ are equal. In particular, the $L^2$-topological torsion of $\bar{M}$ is proportional to the hyperbolic volume of $M$, with a constant of proportionality which depends only on the dimension and which is known to be nonzero in dimension 3, 5 and 7. In dimension 3 this proves the conjecture Of Lott and Lueck which gives a complete calculation of the $L^2$-topological torsion of compact $L^2$-acyclic 3-manifolds which admit a geometric torus-decomposition. In an appendix we give a counterexample to an extension of the Cheeger-Mueller theorem to manifolds with boundary: if the metric is not a product near the boundary, in general analytic and topological torsion are not equal, even if the Euler characteristic of the boundary vanishes. Keywords: L^2-torsion, hyperbolic manifolds, 3-manifoldsComment: 42 pages, AMS-Latex2e V2: identical with published version, in particular including an additional appendix with examples for non-trivial anomaly for analytic torsion on manifolds with boundar

Polyharmonic functions for finite graphs and Markov chains

On a finite graph with a chosen partition of the vertex set into interior and boundary vertices, a $\lambda$-polyharmonic function is a complex function $f$ on the vertex set which satisfies $(\lambda \cdot I - P)^n f(x) = 0$ at each interior vertex. Here, $P$ may be the normalised adjaceny matrix, but more generally, we consider the transition matrix $P$ of an arbitrary Markov chain to which the (oriented) graph structure is adapted. After describing these `global' polyharmonic functions, we turn to solving the Riquier problem, where $n$ boundary functions are preassigned and a corresponding `tower' of $n$ successive Dirichlet type problems are solved. The resulting unique solution will be polyharmonic only at those points which have distance at least $n$ from the boundary. Finally, we compare these results with those concerning infinite trees with the end boundary, as studied by Cohen, Colonnna, Gowrisankaran and Singman, and more recently, by Picardello and Woess

An Application of the Feferman-Vaught Theorem to Automata and Logics for<br> Words over an Infinite Alphabet

We show that a special case of the Feferman-Vaught composition theorem gives rise to a natural notion of automata for finite words over an infinite alphabet, with good closure and decidability properties, as well as several logical characterizations. We also consider a slight extension of the Feferman-Vaught formalism which allows to express more relations between component values (such as equality), and prove related decidability results. From this result we get new classes of decidable logics for words over an infinite alphabet.Comment: 24 page