Dawid Kielak, Robert Kropholler, Gareth Wilkes
We study $\ell^2$ Betti numbers, coherence, and virtual fibring of random groups in the few-relator model. In particular, random groups with negative Euler characteristic are coherent, have $\ell^2$ homology concentrated in dimension 1, and embed in a virtually free-by-cyclic group with high probability. Similar results are shown with positive probability in the zero Euler characteristic case.
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This article presents a method for proving upper bounds for the first $\ell^2$-Betti number of groups using only the geometry of the Cayley graph. As an application we prove that Burnside groups of large prime exponent have vanishing first $\ell^2$-Betti number. Our approach extends to generalizations of $\ell^2$-Betti numbers, that are defined using characters. We illustrate this flexibility by generalizing results of Thom-Peterson on q-normal subgroups to this setting.
We generalise results of Thomas, Allcock, Thom-Petersen, and Kar-Niblo to the first $\ell^2$-Betti number of quotients of certain groups acting on trees by subgroups with free actions on the edge sets of the graphs.
$\ell^2$-Betti numbers and $Σ$-invariants of graph products of cyclic groups and their automorphisms
本文研究循环群的图积的$\ell^2$-Betti数与$\Sigma$-不变量，将其与右角Artin群关联，并在有限阶顶点结构特定情形下给出显式公式，同时计算自同构群的相关不变量。
We explore the Mayer-Vietoris sequence developed by Chiswell for the fundamental group of a graph of groups when vertex groups satisfy some vanishing assumption on the first cohomology (e.g. property (T), or vanishing of the first $\ell^2$-Betti number). We characterize the vanishing of first reduced cohomology of unitary representations when vertex stabilizer have property (T). We find necessary and sufficient conditions for the vanishing of the first $\ell^2$-Betti number. We also study the associated Haagerup cocycle and show that it vanishes in first reduced cohomology precisely when the a…
The purpose of this note is to introduce a trick which relates the (non)-vanishing of the top-dimensional $\ell^2$-Betti numbers of actions with that of sub-actions. We provide three different types of applications: we prove that the $\ell^2$-Betti numbers of Aut($F_n$) and Out($F_n$) (and of their Torelli subgroups) do not vanish in degree equal to their virtual cohomological dimension, we prove that the subgroups of the 3-manifold groups have vanishing $\ell^2$-Betti numbers in degree 3 and 2 and we prove for instance that $F_2^d \times Z$ has ergodic dimension $d + 1$.
Kazhdan groups of dimension $16$ with prescribed second $\ell^2$-Betti number
研究构建了具有性质(T)的简单稀疏超几何群，其理性上同调维数为16，且第二个ℓ²-贝蒂数可预设为任意正实数。同时，还构建了具有性质(T)的超几何群，其第二个ℓ²-贝蒂数可预设为任意非负有理数。
Hereditarily just-infinite torsion groups with positive first $\ell^2$-Betti number
本文提出一种基于 profinite 视角的新方法构造有限生成、剩余有限的无限挠群，首次给出具有正第一 ℓ²-Betti 数的遗传 just-infinite 群例子，并证明其正规子群增长为多项式，回答了 Barnea 和 Schlage-Puchta 的问题。
We provide a proof that the vanishing of $\ell^2$-Betti numbers of unimodular locally compact second countable groups is an invariant of coarse equivalence.
Vanishing of $\ell^2$-Betti numbers for inner amenable groups
该研究解决了内顺从群领域的公开问题，证明可数内顺从群所有次数的$\u2113^2$-贝蒂数均消失。
In this paper, we show that certain families with relative property (T) have trivial first $\ell^2$-Betti number. We apply this to the elementary matrix group $\EL_n(\R)$ where $\R$ is any countable unital ring of characteristic 0.
We show that for a countable exact group, having positive first $\ell^2$-Betti number implies proper proximality in this sense of \cite{BoIoPe21}. This is achieved by showing a cocycle superrigidty result for Bernoulli shifts of non-properly proximal groups. We also obtain that Bernoulli shifts of countable, nonamenable, i.c.c., exact, non-properly proximal groups are OE-superrigid.
We prove that twisted $\ell^2$-Betti numbers of locally indicable groups are equal to the usual $\ell^2$-Betti numbers rescaled by the dimension of the twisting representation; this answers a question of Lück for this class of groups. It also leads to two formulae: given a fibration $E$ with base space $B$ having locally indicable fundamental group, and with a simply-connected fibre $F$, the first formula bounds $\ell^2$-Betti numbers $b_i^{(2)}(E)$ of $E$ in terms of $\ell^2$-Betti numbers of $B$ and usual Betti numbers of $F$; the second formula computes $b_i^{(2)}(E)$ exactly in terms of th…
On the $\ell^2$-Betti numbers and algebraic fibring of the (outer) automorphism group of a right-angled Artin group
本文研究了右角Artin群的$\ell^2$-Betti数及外自同构群的代数纤维化特性，揭示了其非零条件与虚拟纤维化的关系。
In [BV12] we have proven that, for all hyperbolic groups and for all non-trivial free products $Γ$, the left-right wreath product group $G:=(Z/2Z)^{(Γ)} \rtimes (Γ\times Γ)$ is W$^*$-superrigid. In this paper, we extend this result to other classes of countable groups. More precisely, we prove that for weakly amenable groups $Γ$ having positive first $\ell^2$-Betti number, the same wreath product $G$ is W$^*$-superrigid.
A note on normal generation and the first $\ell^2$-betti number
本文构造了满足特定$\ell^2$-Betti数与正规秩关系的可数无挠群，证伪了Osin和Thom关于无挠离散群第一$\ell^2$-Betti数上界的猜想。
We prove that every finitely presented group with positive first $\ell^2$-Betti number that virtually surjects onto $\mathbb Z$ is acylindrically hyperbolic. In particular, this implies acylindrical hyperbolicity of finitely presented residually finite groups with positive first $\ell^2$-Betti number as well as groups of deficiency at least $2$.
We introduce higher-dimensional analogs of Kazhdan projections in matrix algebras over group $C^*$-algebras and Roe algebras. These projections are constructed in the framework of cohomology with coefficients in unitary representations and in certain cases give rise to non-trivial $K$-theory classes. We apply the higher Kazhdan projections to establish a relation between $\ell_2$-Betti numbers of a group and surjectivity of different Baum-Connes type assembly maps.
Higher Kazhdan projections and delocalised $\ell^ 2$-Betti numbers
本文对特定自由积群和直积群的高阶Kazhdan投影的K-类给出了显式描述，并利用该描述计算了Lott的离域$\ell^2$-Betti数，首次得到了无限群的非零结果。
Higher Kazhdan projections and delocalized $\ell^2$-Betti numbers for an amalgamated product group
本文为混合积群ℤₘ*_(ℤ_d)ℤₙ的高阶Kazhdan投影建立K-理论类显式表达式，推广了Pooya和Wang关于自由积群的结果，并应用该表达式得到SL(2,ℤ)离域ℓ²-Betti数的非零结果。
The \emph{normal rank} of a group is the minimal number of elements whose normal closure coincides with the group. We study the relation between the normal rank of a group and its first $\ell^2$-Betti number and conjecture that inequality $β_1^{(2)}(G)$ does not exceed normal rank minus 1 for torsion free groups. The conjecture is proved for limits of left-orderable amenable groups. On the other hand, for every $n\ge 2$ and every $\e>0$, we give an example of a simple group $Q$ (with torsion) such that $β_1^{(2)}(Q) \geq n-1-ε$. These groups also provide examples of simple groups of rank exact…
Let $n\ge 3$, and let $\mathrm{Out}(W_n)$ be the outer automorphism group of a free Coxeter group $W_n$ of rank $n$. We study the growth of the dimension of the homology groups (with coefficients in any field $\mathbb{K}$) along Farber sequences of finite-index subgroups of $\mathrm{Out}(W_n)$. We show that, in all degrees up to $\lfloor\frac{n}{2}\rfloor-1$, these Betti numbers grow sublinearly in the index of the subgroup. When $\mathbb{K}=\mathbb{Q}$, through Lück's approximation theorem, this implies that all $\ell^2$-Betti numbers of $\mathrm{Out}(W_n)$ vanish up to degree $\lfloor\frac{n…
In 1987, Ornstein and Weiss discovered that the Bernoulli $2$-shift over the rank two free group factors onto the seemingly larger Bernoulli $4$-shift. With the recent creation of an entropy theory for actions of sofic groups (in particular free groups), their example shows the surprising fact that entropy can increase under factor maps. In order to better understand this phenomenon, we study a natural generalization of the Ornstein--Weiss map for countable groups. We relate the increase in entropy to the cost and to the first $\ell^2$-Betti number of the group. More generally, we study coboun…
In this note we refine examples by Aka from arithmetic to S-arithmetic groups to show that the vanishing of the $i$-th $\ell^2$-Betti number is not a profinite invariant for all $i \ge 2$.
Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. On a Cayley graph of a group, we show that they are related to the first $\ell^2$-Betti number of the group. Our main aim, however, is to present the basic elements of a higher-dimensional analogue on finite and infinite CW-complexes, which relate to the higher $\ell^2$-Betti numbers. One consequence is a uniform isoperimetric inequality extending work of Lyons, Pichot, and Vassout. We also present an enumeration similar to recent work of Duval, Klivans, and Martin.
L^2-Betti numbers arising from the lamplighter group
本文利用交叉积代数构造公式，计算灯夫群代数及广义里程计代数的无理ℓ²-Betti数，扩展了Grabowski的结果，并完全刻画了里程计代数的ℓ²-Betti数集合。
It is shown that $2β_1(\G)\leq h(\G)$ for any countable group $\G$, where $β_1(\G)$ is the first $\ell^2$-Betti number and $h(\G)$ the uniform isoperimetric constant. In particular, a countable group with non-vanishing first $\ell^2$-Betti number is uniformly non-amenable. We then define isoperimetric constants in the framework of measured equivalence relations. For an ergodic measured equivalence relation $R$ of type $\IIi$, the uniform isoperimetric constant $h(R)$ of $R$ is invariant under orbit equivalence and satisfies $$ 2β_1(R)\leq 2C(R)-2\leq h(R), $$ where $β_1(\R)$ is the first $\ell…
Coherent RFRS groups
本文研究上同调维数至多2的有限生成虚拟RFRS群的凝聚性等价条件，据此刻画二维Coxeter群等类群的凝聚性并证实相关猜想，附录中解决了立方化情形下的相关公开问题。
We examine a condition on a simply connected 2-complex X ensuring that groups acting properly on X are coherent. This extends earlier work on 2-complexes with negative sectional curvature which covers the case that G acts freely. Our extension of these results involves a generalization of the notion of sectional curvature, an extension of the combinatorial Gauss-Bonnet theorem to complexes of groups, and surprisingly requires the use of L^2-Betti numbers. We also prove local quasiconvexity of G under the additional assumption that X is CAT(0) space.
The generalized (ordered) configuration spaces associated to a topological space $X$ are the spaces $Δ_{\leq\ell}X^{m}:=\{(x_1,\ldots,x_{m})\in X^{m}\mid\#\{x_1,\ldots,x_{m}\}\leq \ell\}$ and $Δ_{\ell}X^{m}:=Δ_{\leq\ell}X^{m}\setminus Δ_{\leq\ell-1}$. They are equipped with the action of the symmetric group $S_m$ permuting coordinates. When $X$ has no interior cohomology (i.e. is $i$-acyclic) we are able to compute explicitly the character formula of $S_m$ acting on the cohomology of these spaces, and if $X$ is furthermore a connected and oriented pseudomanifold of dimension $\geq2$ we general…
An $\ell^2$ Obstruction for Elementary Embeddings of Hyperbolic Groups
本文研究了超几何群在第一阶逻辑下嵌入时 $\ell^2$ 零数的单调性，证明了非阿贝尔proper子群在无 torsion 超几何群中的元素嵌入下严格递增，并进一步展示了任意有限生成群的存在性嵌入的单调性。
BNSR invariants and $\ell^2$-homology
本研究在适当有限性条件下证明群的n次ℓ²-Betti数非零时其n次Q上BNSR不变量为空，将结果应用于非球面Kähler流形问题、Singer猜想验证及多自由群与多曲面群的BNSR不变量计算。
Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a cubical complex Σ_L on which W_L acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of Σ_L is L and (2) Σ_L is contractible. It follows that if L is a triangulation of S^{n-1}, then Σ_L is a contractible n-manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced L^2-homology except in the middle dimension) in the case of Σ_L where L is a triangulation of S^{n-1}. The program succeeds when n < 5. This implies the Charney…
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