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We consider the problem whether for a group [Formula: see text] there exists a constant [Formula: see text] such that for any [Formula: see text]-matrix [Formula: see text] over the integral group ring [Formula: see text] the Fuglede–Kadison determinant of the [Formula: see text]-equivariant bounded operator [Formula: see text] given by right multiplication with [Formula: see text] is either one or greater or equal to [Formula: see text]. If [Formula: see text] is the infinite cyclic group and we consider only [Formula: see text], this is precisely Lehmer’s problem.

In this paper give a survey about L^2-invariants focusing on 3-manifolds., Comment: 38 pages. This is the final version and has been accepted for publication in the Bulletin of the London Mathematical Society

Let M be an oriented irreducible 3-manifold with infinite fundamental group and empty or toroidal boundary. Consider any element \phi in the first cohomology of M with integral coefficients. Then one can define the \phi-twisted L^2-torsion function of the universal covering which is a function from the set of positive real numbers to the set of real numbers. By earlier work of the second author and Schick the evaluation at t=1 determines the volume. In this paper we show that its degree, which is a number extracted from its asymptotic behavior at 0 and at infinity, agrees with the Thurston norm of \phi., Comment: 22 pages. v2: minor changes

We assign to a finite $CW$-complex and an element in its first cohomology group a twisted version of the $L^2$-Euler characteristic and study its main properties. In the case of an irreducible orientable $3$-manifold with empty or toroidal boundary and infinite fundamental group we identify it with the Thurston norm. We will use the $L^2$-Euler characteristic to address the problem whether the existence of a map inducing an epimorphism on fundamental groups implies an inequality of the Thurston norms., Comment: 44 pages, updated references, incorporated comments by the referee

Given an L2-acyclic connected finite CW-complex, we define its universal L2-torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead group Whw(G). We study its main properties such as homotopy invariance, sum formula, product formula and Poincare duality. Under certain assumptions, we can specify certain homomorphisms from the weak Whitehead group Whw(G) to abelian groups such as the real numbers or the Grothendieck group of integral polytopes, and the image of the universal L2-torsion can be identified with many invariants such as the L2-torsion, the L2-torsion function, twisted L2-Euler characteristics and, in the case of a 3-manifold, the dual Thurston norm polytope.

We formulate and prove a version of the Segal conjecture for infinite groups. For finite groups it reduces to the original version. The condition that [math] is finite is replaced in our setting by the assumption that there exists a finite model for the classifying space [math] for proper actions. This assumption is satisfied for instance for word hyperbolic groups or cocompact discrete subgroups of Lie groups with finitely many path components. As a consequence we get for such groups [math] that the zeroth stable cohomotopy of the classifying space [math] is isomorphic to the [math] –adic completion of the ring given by the zeroth equivariant stable cohomotopy of [math] for [math] the augmentation ideal.

We compute the topological simple structure set of closed manifolds which occur as total spaces of flat bundles over lens spaces S^l/(Z/p) with fiber an n-dimensjional torus T^n for an odd prime p and l greater or equal to 3, provided that the induced Z/p-action on pi_1(T^n) = Z^n is free outside the origin. To the best of our knowledge this is the first computation of the structure set of a topological manifold whose fundamental group is not obtained from torsionfree and finite groups using amalgamated and HNN-extensions. We give a collection of classical surgery invariants such as splitting obstructions and rho-invariants which decide whether a simple homotopy equivalence from a closed topological manifold to M is homotopic to a homeomorphism., 40 pages, to appear in Communications on Pure and Applied Mathematics

We compute the group homology, the algebraic $K$- and $L$-groups, and the topological $K$-groups of right-angled Artin groups, right-angled Coxeter groups, and more generally, graph products., 32 pages

International audience; Deforestation is a permanent land use change, caused by the direct human-induced conversion of forest land to other land uses. Land cover changes such as forest clearings due to harvesting as part of sustainable forest management, or due to natural disturbances, are not deforestation, as they are usually temporary and followed by forest re-growth. Current pixel based change detection techniques are sensitive to tree cover change and specific sensor characteristics and are, therefore, ill-suited for automated land use change classification. With a limited number of temporal observations, Geographic Object Based Image Analysis (GEOBIA) has been widely used for deforestation mapping in projects such as REDD+, but has not yet been implemented within a data cube architecture, such as the Open Data Cube (ODC). The purpose of this research project is to use analysis ready data (ARD) acquired by the Landsat Program and Sentinels 1 and 2 over Canada, to detect deforestation using the ODC with a GEOBIA data abstraction layer. Using both SAR and optical observations, the Continuous Change Detection and Classification (CCDC) algorithm is then applied on a per segment basis to characterize historic land use versus current land use, and to identify the resulting change date. To generate a quality layer for the ODC, a pixel based spectral pre-classification delineating spectral classes such as cloud, haze, shadow, water, snow, dark bare soil, bright bare soil, herbaceous vegetation and woody vegetation, is applied to every image. A reference object layer is generated from segmented Sentinel 2 10m resolution imagery acquired in summer of 2017 by incorporating multi-date acquisitions until all segments have been formed from clean data. For the analysis, only segments are used which do not contain any cloud, shadow, haze, or snow pixels. In the time series for each segment, when a discontinuity in pixel based class mixtures is identified, the segment is split along the discontinuity boundary. In the ODC, once segments have been established across the full time series, and spectral/textural features have been calculated for each segment, CCDC is used to characterize the phenology of each segment. Abrupt changes in phenological behavior can be identified with given break points. When the land cover classes on each side of the temporal divide are identified, they can be used to determine if the non-forest land cover reverts back to forest or not. This will distinguish between land cover change and land use change. Only a change in land use is flagged as true change, and this can then be used to assist in deforestation monitoring. The study area is situated in northeastern Alberta, Canada, and is historically subjected to frequent deforestation activities, mainly attributed to oil sands mining and fracking. Using manual image interpretation techniques, the Canadian Forest Service has recorded deforestation in a systematic sampling design at frequent intervals for the small events, and by a complete coverage for large events. This highly accurate reference data is used to validate the performance of automatic deforestation mapping techniques developed in this study.

We show that the $L^2$-Alexander torsion of a 3-manifold is symmetric. This can be viewed as a generalization of the symmetry of the Alexander polynomial of a knot., 14 pages. arXiv admin note: text overlap with arXiv:1107.3002

Let Wh^w(G) be the K_1-group of square matrices over the integral group ring ZG which are not necessarily invertible but induce weak isomorphisms after passing to Hilbert space completions. Let D(G) be the division closure of ZG in the algebra U(G) of operators affiliated to the group von Neumann algebra. Let C be the smallest class of groups which contains all free groups and is closed under directed unions and extensions with elementary amenable quotients. Let G be a torsionfree group which belongs to C. Then we prove that Wh^w(G) is isomorphic to K_1(D(G)). Furthermore we show that D(G) is a skew field and henc K_1(\D(G)) is the abelianization of the multiplicative group of units in D(G)., Comment: 18 pages, final version, to appear in Annals of K-theory

We prove the Farrell-Jones Conjecture for (non-connective) $A$-theory with coefficients and finite wreath products for hyperbolic groups, CAT(0)-groups, cocompact lattices in almost connected Lie groups and fundamental groups of manifolds of dimension less or equal to three. Moreover, we prove inheritance properties such as passing to subgroups, colimits of direct systems of groups, finite direct products and finite free products. These results hold also for Whitehead spectra and spectra of stable pseudo-isotopies in the topological, piecewise linear and smooth category., Comment: 49 pages; improved exposition

The aim of this note is to introduce L 2 -Alexander torsions for 3-manifolds (which are generalizations of the usual Alexander polynomial and also of the L 2 -Alexander invariant defined by Li and Zhang [7] ) and to report on calculations for graph manifolds and fibered 3-manifolds. We further announce that given any irreducible 3-manifold, there exists a coefficient system such that the corresponding L 2 -Alexander torsion detects the Thurston norm. Finally we also state a symmetry formula.

Let X be a finite graph. Let | V | be the number of its vertices and d be its degree. Denote by F 1 ( X ) its first spectral density function which counts the number of eigenvalues ≤ λ 2 of the associated Laplace operator. We provide an elementary proof for the estimate F 1 ( X ) ( λ ) − F 1 ( X ) ( 0 ) ≤ 2 ⋅ ( | V | − 1 ) ⋅ d ⋅ λ for 0 ≤ λ 1 which has already been proved by Friedman (1996) [3] before. We explain how this gives evidence for conjectures about approximating Fuglede–Kadison determinants and L 2 -torsion.

We study the asymptotic growth of Betti numbers in tower of finite covers and provide simple proofs of approximation results, which were previously obtained by Calegari-Emerton, in the generality of arbitrary p-adic analytic towers of covers. Further, we also obtain partial results about arbitrary pro-$p$ towers., 15 pages, the previous version contained an in the proof of Theorem 2.1 that is now corrected. This is the last version, it will appear in GGD

We prove the K- and L-theoretic Farrell-Jones Conjecture (with coefficients in additive categories) for GL n (Z).

In this paper we consider the asymptotic behavior of invariants such as Betti numbers, minimal numbers of generators of singular homology, the order of the torsion subgroup of singular homology, and torsion invariants. We will show that all these vanish in the limit if the CW-complex under consideration fibers in a specific way. In particular we will show that all these vanish in the limit if one considers an aspherical closed manifold which admits a non-trivial S1-action or whose fundamental group contains an infinite normal elementary amenable subgroup. By considering classifying spaces we also get results for groups.