Sectional rank and Cohomology II

A sectional characterization of the Dade group

Let $k$ be a field of characteristic $p$, let $P$ be a finite $p$- group, where $p$ is an odd prime, and let $D(P)$ be the Dade group of endo-permutation $kP$-modules. It is known that $D(P)$ is detected via deflation--restriction by the family of all sections of $P$ which are elementary abelian of rank $\leq2$. In this paper, we improve this result by characterizing $D(P)$ as the limit (with respect to deflation--restriction maps and conjugation maps) of all groups $D(T/S)$ where $T/S$ runs through all sections of $P$ which are either elementary abelian of rank $\leq3$ or extraspecial of orde…

The cohomology rings of rank 3 simple groups are Cohen-Macaulay

In this paper we show that the mod 2 cohomology ring of any finite simple group of rank 3 or less (at the prime 2) must be Cohen-Macaulay.

On the low dimensional cohomology groups of the IA-automorphism group of a free group of rank three

In this paper we study the structure of the rational cohomology groups of the IA-automorphism group $\mathrm{IA}_3$ of a free group of rank three by using combinatorial group theory and representation theory. In particular, we detect non-trivial irreducible component in the second cohomology group of $\mathrm{IA}_3$, which does not contained in the image of the cup product map of the first cohomology groups. We also show that the image of the triple cup product map of the first cohomology groups in the third cohomology group is trivial. As a corollary, we obtain that the fourth term of the low…

De Rham L^p-上同调对于高秩空间与群：临界指数与刚性

De Rham $L^p$-Cohomology For Higher Rank Spaces And Groups: Critical Exponents And Rigidity

研究高秩李群和相关流形的L^p上同调临界指数及刚性特性，通过谱序列论证揭示非正曲率可解李群的连续准等距类。

The cohomology of virtually torsion-free solvable groups of finite rank

Assume that $G$ is a virtually torsion-free solvable group of finite rank and $A$ a $\mathbb ZG$-module whose underlying abelian group is torsion-free and has finite rank. We stipulate a condition on $A$ that ensures that $H^n(G,A)$ and $H_n(G,A)$ are finite for all $n\geq 0$. Using this property for cohomology in dimension two, we deduce two results concerning the presence of near supplements and complements in solvable groups of finite rank. As an application of our near-supplement theorem, we obtain a new result regarding the homological dimension of solvable groups.

Weak notions of normality and vanishing up to rank in L2-cohomology

We study vanishing results for L2-cohomology of countable groups under the presence of subgroups that satisfy some weak normality condition. As a consequence we show that the L2-Betti numbers of SL(n,R) for any infinite integral domain R vanish below degree n-1. We also give a uniform proof for the vanishing of L2-Betti numbers of Thompson's groups F and T.

大多数秩至少为3的半单群的第二$L^p$-上同调群的消失

Vanishing of the second $L^p$-cohomology group for most semisimple groups of rank at least 3

本文证明大多数秩至少3的半单代数群的第二$L^p$-上同调群消失，方法结合谱序列、Mautner现象与组合逐例分析，支持Gromov猜想。

A characterization of higher rank symmetric spaces via bounded cohomology

Let $M$ be complete nonpositively curved Riemannian manifold of finite volume whose fundamental group $Γ$ does not contain a finite index subgroup which is a product of infinite groups. We show that the universal cover $\tilde M$ is a higher rank symmetric space iff $H^2_b(M;\R)\to H^2(M;\R)$ is injective (and otherwise the kernel is infinite-dimensional). This is the converse of a theorem of Burger-Monod. The proof uses the celebrated Rank Rigidity Theorem, as well as a new construction of quasi-homomorphisms on groups that act on CAT(0) spaces and contain rank 1 elements.

Mod 2 cohomology of 2-local finite groups of low rank

We determine the mod $2$ cohomology over the Steenrod algebra of the classifying spaces of the free loop groups $LG$ for compact groups $G=Spin(7)$, $Spin(8)$, $Spin(9)$, and $F_4$. Then, we show that they are isomorphic as algebras over the Steenrod algebra to the mod $2$ cohomology of the corresponding Chevalley groups of type $G(q)$, where $q$ is an odd prime power. In a similar manner, we compute the cohomology of the free loop space over $BDI(4)$ and show that it is isomorphic to that of $BSol(q)$ as algebras over the Steenrod algebra.

Complexity and cohomology of cohomological Mackey functors

Let $k$ be a field of characteristic $p>0$. Call a finite group $G$ a poco group over $k$ if any finitely generated cohomological Mackey functor for $G$ over $k$ has polynomial growth. The main result of this paper is that $G$ is a poco group over $k$ if and only if the Sylow $p$-subgroups of $G$ are cyclic, when $p>2$, or have sectional rank at most 2, when $p=2$. A major step in the proof is the case where $G$ is an elementary abelian $p$-group. In particular, when $p=2$, all the extension groups between simple functors can be determined completely, using a presentation of the graded algebra…

Torsion cohomology for solvable groups of finite rank

We define a class $\mathcal{U}$ of solvable groups of finite abelian section rank which includes all such groups that are virtually torsion-free as well as those that are finitely generated. Assume that $G$ is a group in $\mathcal{U}$ and $A$ a $\mathbb ZG$-module. If $A$ is $\mathbb Z$-torsion-free and has finite $\mathbb Z$-rank, we stipulate a condition on $A$ that guarantees that $H^n(G,A)$ and $H_n(G,A)$ must be finite for $n\geq 0$. Moreover, if the underlying abelian group of $A$ is a Černikov group, we identify a similar condition on $A$ that ensures that $H^n(G,A)$ must be a Černikov …

On groups of finite upper rank

The `upper rank' of a group is the supremum of the (Prüfer) ranks of its finite quotients, and for a prime $p$, the `upper $p$-rank' is the supremum of the sectional $p$-ranks of those quotients. The former is finite if and only if the latter are finitely bounded as $p$ ranges over all primes (a deep fact). Here we discuss the question: if the upper $p$-ranks of a finitely generated group $G$ are all finite, are they necessarily bounded? The case where $G$ is a soluble group is still an open problem.

Cohomology of $Aut(F_n)$ in the p-rank two case

For odd primes p, we examine $\hat H^*(Aut(F_{2(p-1)}); \Z_{(p)})$, the Farrell cohomology of the group of automorphisms of a free group $F_{2(p-1)}$ on $2(p-1)$ generators, with coefficients in the integers localized at the prime $(p) \subset \Z$. This extends results by Glover and Mislin, whose calculations yield $\hat H^*(Aut(F_n); \Z_{(p)})$ for $n \in \{p-1,p\}$ and is concurrent with work by Chen where he calculates $\hat H^*(Aut(F_n); \Z_{(p)})$ for $n \in \{p+1,p+2\}$. The main tools used are Ken Brown's ``normalizer spectral sequence'', a modification of Krstic and Vogtmann's proof of…

算术群的Banach表示的固定点性质与上同调

Fixed point properties and cohomology of Banach representations of arithmetic groups

本文研究半单群格的固定点定理，通过证明高秩单群格在无不变向量时秩以下$L^p$上同调消失，推出低维无环复形作用有有限轨道，并证明Farb与Gromov的猜想。

The cohomology of semi-simple Lie groups, viewed from infinity

We prove that the cohomology of semi-simple Lie groups admits boundary values, which are measurable cocycles on the Furstenberg boundary. This generalises known invariants such as the Maslov index on Shilov boundaries, the Euler class on projective space, or the hyperbolic ideal volume on spheres. In rank one, this leads to an isomorphism between the cohomology of the group and of this boundary model. In higher rank, additional classes appear, which we determine completely.

Non-vanishing for group $L^p$-cohomology of solvable and semisimple Lie groups

We obtain non-vanishing of group $L^p$-cohomology of Lie groups for $p$ large and when the degree is equal to the rank of the group. This applies both to semisimple and to some suitable solvable groups. In particular, it confirms that Gromov's question on vanishing below the rank is formulated optimally. To achieve this, some complementary vanishings are combined with the use of spectral sequences. To deduce the semisimple case from the solvable one, we also need comparison results between various theories for $L^p$-cohomology, allowing the use of quasi-isometry invariance.

有界上同调不是profinite不变量

Bounded cohomology is not a profinite invariant

本文证明了有界上同调并非profinite不变量，通过构造具有同构profinite完备化但实二阶有界上同调不同的剩余有限群对，并利用Galois上同调给出了允许此类例子的所有高秩Lie群的完整列表。

低秩整数特殊线性群的非消失酉上同调

Non-vanishing unitary cohomology of low-rank integral special linear groups

本文构造了低秩整数特殊线性群的正交表示，证明其上同调非平凡，从而表明该群不满足更高版本的Kazhdan性质T。