Graphs of groups and the Atiyah conjecture for one-relator groups

Proof of Atiyah's conjecture for two special types of configurations

We prove Atiyah's conjecture for two special types of configurations of N points in the three-dimensional Euclidean space. For one of these types, it is shown that the stronger conjecture of Atiyah and Sutcliffe is valid.

单关系子层级

One-relator hierarchies

本文通过改进经典的 Magnus--Moldavanskii 层级并引入 Z-稳定 HNN-扩张等概念，证明了具有负浸入的单关系群是双曲且几乎特殊的，解决了 Louder 和 Wilton 的猜想及 Baumslag 的问题。

The surjectivity problem for one-generator,, one-relator extensions of torsion-free groups

We use Klyachko's methods to prove that the natural map G to G-hat, where G is a torsion-free group and G-hat is obtained by adding a new generator t and a new relator w, is surjective only if w is conjugate to gt or gt^{-1} for some g in G. This solves a special case of the surjectivity problem for group extensions, raised by Cohen [Whitehead torsion, group extensions, and Zeeman's conjecture in high dimensions, Topology, 16 (1977) 79--88].

Verification of Atiyah's conjecture for some nonplanar configurations with dihedral symmetry

Atiyah's conjecture concerning configurations of N points in the Euclidean three-space is verified for the following nonplanar configurations: The first m points lie on a line L and the remaining n=N-m (>2) points are the vertices of a regular n-gon whose plane is perpendicular to L and whose centroid is on L.

Atiyah-Floer Conjecture: a Formulation, a Strategy to Prove and Generalizations

Around 1988, Floer introduced two important theories: instanton Floer homology as invariants of 3-manifolds and Lagrangian Floer homology as invariants of pairs of Lagrangians in symplectic manifolds. Soon after that, Atiyah conjectured that the two theories should be related to each other and Lagrangian Floer homology of certain Lagrangians in the moduli space of flat connections on Riemann surfaces should recover instanton Floer homology. However, the space of flat connections on a Riemann surface is singular and the first step to address this conjecture is to make sense of Lagrangian Floer …

Integrality of L2-Betti numbers

The Atiyah conjecture predicts that the L2-Betti numbers of a finite CW-complex with torsion-free fundamental group are integers. We show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of directed systems of groups for which it is true. As a corollary it holds for residually torsion-free solvable groups, e.g. for pure braid groups or for positive 1-relator groups with torsion free abelianization. Putting everything together we establish a new class of groups for which the Atiyah conjecture holds, which contains all free groups and in par…

Negative immersions for one-relator groups

We prove a freeness theorem for low-rank subgroups of one-relator groups. Let $F$ be a free group, and let $w\in F$ be a non-primitive element. The primitivity rank of $w$, $π(w)$, is the smallest rank of a subgroup of $F$ containing $w$ as an imprimitive element. Then any subgroup of the one-relator group $G=F/\langle\langle w\rangle\rangle$ generated by fewer than $π(w)$ elements is free. In particular, if $π(w)>2$ then $G$ doesn't contain any Baumslag--Solitar groups. The hypothesis that $π(w)>2$ implies that the presentation complex $X$ of the one-relator group $G$ has negative immersions:…

Congruence subgroups and the Atiyah conjecture

Let A denote the algebraic closure of the rationals Q in the complex numbers C. Suppose G is a torsion-free group which contains a congruence subgroup as a normal subgroup of finite index and denote by U(G) the C-algebra of closed densely defined unbounded operators affiliated to the group von Neumann algebra. We prove that there exists a division ring D(G) such that A[G] < D(G) < U(G). This establishes some versions of the Atiyah conjecture for the group G.

Characteristics of graph braid groups

We give formulae for the first homology of the $n$-braid group and the pure 2-braid group over a finite graph in terms of graph theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the $n$-braid group over the graph is torsion-free and the conjectures about the first homology of the pure 2-braid groups over graphs in \cite{FH} can be verified. We discover more characteristics of graph braid groups: the $n$-braid group over a planar graph and the pure 2-braid group over any graph have a presentation whose relators are words of commutators, and …

On the Atiyah problem on hyperbolic configurations of four points

Given a configuration $\mathbf{x}$ of $n$ distinct points in hyperbolic $3$-space $H^3$, Michael Atiyah associated $n$ polynomials $p_1,\ldots,p_n$ of a variable $t \in \mathbb{C}P^1$, of degree $n-1$, and conjectured that they are linearly independent over $\mathbb{C}$, no matter which configuration $\mathbf{x}$ one starts with. We prove this conjecture for $n=4$ in two cases: in case the $4$ points are non-coplanar, and in case one of the points lies in the hyperbolic convex hull of the other three.

Weak $\mathcal Z$-structures and one-relator groups

Motivated by the notion of boundary for hyperbolic and $CAT(0)$ groups, M. Bestvina in "Local Homology Properties of Boundaries of Groups" introduced the notion of a (weak) $\mathcal Z$-structure and (weak) $\mathcal Z$-boundary for a group $G$ of type $\mathcal F$ (i.e., having a finite $K(G,1)$ complex), with implications concerning the Novikov conjecture for $G$. Since then, some classes of groups have been shown to admit a weak $\mathcal Z$-structure (see "Weak $\mathcal Z$-structures for some classes of groups" by C.R. Guilbault for example), but the question whether or not every group of…

凯尔韦尔猜想与曲面的最小复杂度

The Kervaire conjecture and the minimal complexity of surfaces

本研究引入稳定交换子长度新方法，研究单关系商非平凡性问题，证明特定HNN扩张中曲面复杂度不低于边界，得到HNN扩张的自由性定理，给出无挠群凯尔韦尔猜想新证明并推广Klyachko-Lurye定理。

Approximating L2-invariants, and the Atiyah conjecture

Let G be a torsion free discrete group and let \bar{Q} denote the field of algebraic numbers in C. We prove that \bar{Q}[G] fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups which are residually torsion free elementary amenable or which are residually free. This result implies that there are no non-trivial zero-divisors in C[G]. The statement relies on new approximation results for L2-Betti numbers over \bar{Q}[G], which are the core of the work done in this paper. Another set of results in the paper is concerned with certain numbe…

A Compactness Theorem for $SO(3)$ Anti-Self-Dual Equation with Translation Symmetry

Motivated by the Atiyah-Floer conjecture, we consider $SO(3)$ Santi-self-dual instantons on the product of the real line and a three-manifold with cylindrical end. We prove a Gromov-Uhlenbeck type compactness theorem, namely, any sequence of such instantons with uniform energy bound has a subsequence converging to a type of singular objects which may have both instanton and holomorphic curve components. This result is the first step towards constructing a natural bounding cochain proposed by Fukaya for the $SO(3)$ Atiyah-Floer conjecture.

三维流形群的群环

Group rings of three-manifold groups

该研究针对三维流形基本群的复群环展开，整合已知结论证明了有理上同调维数为2时群环的凝聚性，以及无挠情形下强Atiyah猜想与Kaplansky零因子猜想的成立性。

Helly meets Garside and Artin

A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, additionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in parti…

Thurston 范数、多面体与分裂复杂度

Thurston norm, polytopes and splitting complexity

研究满足强Atiyah猜想的无挠群，将核的第一L²-Betti数延拓为Thurston半范数，证明其由同调群中的多面体诱导，并推广到高阶L²-Betti数；在自由-循环群和三维流形基本群情形给出组合解释，应用于计算BNS不变量。