Jan Essert
We prove homological stability for standard unitary groups over R, C and H and for general linear groups over skew-fields with infinite centre. We focus on the similarities and differences of these proofs. Both proofs are due to Chih-Han Sah (Homology of classical Lie groups made discrete I: Stability theorems and Schur multipliers. Comment. Math. Helv. 61(2), 1986).
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On the Homological Stability of Orthogonal and Spin Groups
本文改进了在特定局部环上正交和旋转变换群的同调稳定性范围。
Unstable algebraic K-theory: homological stability and other observations
本文研究约化Borel-Serre范畴的稳定性性质，证明其同调稳定性优于一般线性群，并为其作为Yuan部分代数K-理论的显式模型提供了依据。
A criterion for slope 1 homological stability
本文针对足够良好的N-分次E2-代数，证明其E1-同调中的对角消失线能导出斜率1的同调稳定性，并给出Kupers-Miller-Patzt结果的整数版本。
Unordered resolutions and homological stability for linear groups
本文通过改进的证明策略研究线性群的同调稳定性，重点探讨了局部化系数与若干猜想的关系。
The problem of homological stability helps us to catch the structure of group homology. We calculate homological stability of special orthogonal groups, and we also calculate the stability of orthogonal groups with determinant-twisted coefficients under a certain good situation. We also get some results about the structure of these homology.
Improved homological stability for certain general linear groups
该研究改进了整数、高斯整数及艾森斯坦整数对应的一般线性群的同调稳定性结果，明确了不同系数下的斜率值。
Associated to every group with a weak spherical Tits system of rank n+1 with an appropriate rank n subgroup, we construct a relative spectral sequence involving group homology of Levi subgroups of both groups. Using the fact that such Levi subgroups frequently split as semidirect products of smaller groups, we prove homological stability results for unitary groups over division rings with infinite centre as well as for special linear and special orthogonal groups over infinite fields.
We prove twisted homological stability with polynomial coefficients for automorphism groups of free nilpotent groups of any given class. These groups interpolate between two extremes for which homological stability was known before, the general linear groups over the integers and the automorphism groups of free groups. The proof presented here uses a general result that applies to arbitrary extensions of groups, and that has other applications as well.
Homological stability for general linear groups over Dedekind domains
该研究针对 Dedekind 整环上有限生成投射模的自同构群，提出了考虑所有稳定化映射的新型同调稳定性定理，并验证了约化 Borel--Serre 空间也具备该稳定性。
Homological stability for automorphisms of symmetric bilinear forms
本文研究了对称双线性形式的自同构的同调稳定性，结合Grothendieck-Witt理论计算，确定了奇正交群在低度数中的稳定上同调结构。
Homological stability for the ribbon Higman--Thompson groups
本文通过引入ribbon Higman-Thompson群的概念，证明了其满足同调稳定性，为大映射类群的稠密子群提供了首个同调稳定性结果。
We formulate a "correct" version of the Quillen conjecture on linear group homology for certain arithmetic rings and provide evidence for the new conjecture. In this way we predict that the linear group homology has a direct summand looking like an unstable form of Milnor K-theory and we call this new theory "homological symbols algebra". As a byproduct we prove the Quillen conjecture in homological degree two for the rank two and the prime 5.
The unimodularity condition for a Poisson structure (ie., a Poisson structure with a trivial modular class) induces a Poincaré duality between its Poisson homology and its Poisson cohomology. Therefore an information about the Poisson homology of this kind Poisson structures induces by duality an information about its Poisson cohomology and vise versa. But this is not longer true in the case of a non trivial modular class. That is the case of Generalized Jacobian Poisson Structures (GJPS). In this paper, we consider certain GJPS in dimension 3 and obtain properties of their Poisson homological…
Homological duality and exact sequences of Hopf algebras
本文研究 Hopf 代数在扩张下同调对偶性质的稳定性，提出并证明了相关正合序列中的保持条件。
We prove a slope 1 stability range for the homology of the symplectic, orthogonal and unitary groups with respect to the hyperbolic form, over any fields other than $F_2$, improving the known range by a factor 2 in the case of finite fields. Our result more generally applies to the automorphism groups of vector spaces equipped with a possibly degenerate form (in the sense of Bak, Tits and Wall). For finite fields of odd characteristic, and more generally fields in which -1 is a sum of two squares, we deduce a stability range for the orthogonal groups with respect to the Euclidean form, and a c…
Homological stability fails for the Cremona groups, yet their stable homology is non-trivial
本文研究克雷莫纳群，发现其同调稳定性不成立，但证明其稳定同调由非平凡无限循环空间表示，为这类群的同调研究提供了新视角。
We prove a version of the Cohen--Lenstra conjecture over function fields (completing the results of our prior paper). This is deduced from two more general theorems, one topological, one arithmetic: We compute the direct limit of homology, over puncture-stabilization, of spaces of maps from a punctured manifold to a fixed target; and we compute the Galois action on the set of stable components of Hurwitz schemes.
Partial bases and homological stability of $\operatorname{GL}_{n}(R)$ revisited
本文针对满足不变基性质等三个条件的有单位元环$R$，推广已有研究成果，证明一般线性群序列$\operatorname{GL}_n(R)$在$\mathbb{Z}[1/2]$系数下具有斜率为1的同调稳定性。
We give a new categorical way to construct the central stability homology of Putman and Sam and explain how it can be used in the context of representation stability and homological stability. In contrast to them, we cover categories with infinite automorphism groups. We also connect central stability homology to Randal-Williams and Wahl's work on homological stability. We also develop a criterion that implies that functors that are polynomial in the sense of Randal-Williams and Wahl are centrally stable in the sense of Putman.
We prove a homological stability theorem for congruence subgroups of symplectic groups. From this theorem, we deduce a generalization of a theorem of Borel showing that certain homology groups of a congruence subgroup do not depend on the level of the congruence subgroup.
Group class operations and homological conditions
本文利用 Kropholler 的 LH 与 Talelli 的 Φ 群类扩充运算，证明大量群同时具有虚拟 Gorenstein 群代数并满足 Moore 关于投射性与相对投射性的猜想。
Scissors automorphism groups I: Homological stability and K-theory
本文证明在温和条件下剪刀自同构群的同调独立于对象，可由剪刀同余 K-理论谱表达，提供群论解释与计算方法，并在经典情形及后续中得出同调计算与结构结果。
In 1972, B. E. Johnson proved that a locally compact group $G$ is amenable if and only if certain Hochschild cohomology groups of its convolution algebra $L^1(G)$ vanish. Similarly, $G$ is compact if and only if $L^1(G)$ is biprojective: In each case, a classical property of $G$ corresponds to a cohomological propety of $L^1(G)$. Starting with the work of Z.-J. Ruan in 1995, it has become apparent that in the non-commutative setting, i.e. when dealing with the Fourier algebra $A(G)$ or the Fourier-Stieltjes algebra $B(G)$, the canonical operator space structure of the algebras under considerat…
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