Questions tagged [group-homology]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
7 votes
1 answer
452 views

Computing homology groups with GAP

I’m studying the homology groups of arithmetic groups such as $SL(5,\mathbb{Z})$. I saw in the answer to this post that we can use GAP to compute some of the homology groups for $SL(3,\mathbb{Z})$. Is ...
Noah B's user avatar
  • 403
2 votes
0 answers
76 views

Reference request: Lower group (co)homology of linear groups

I am finding references about the following general question: What is the group (co)homology $H_{i}(G,\mathbb{Z})$ for a linear group $G$? In my case, I'm particularly interested in the special case ...
hyyyyy's user avatar
  • 233
7 votes
1 answer
339 views

How can I detect the homology image of a unipotent group in the general linear group?

Suppose $n$ is a positive integer greater than 2, and $F$ is an arbitrary field with at least 4 elements. Denote $\text{GL}_n(F)$ the general linear group in the usual sense and $U_n(F)$ the unipotent ...
XYC's user avatar
  • 389
4 votes
1 answer
263 views

The third homology stability of general linear groups over finite fields

Given a finite field $\mathbb{F}$ with $|\mathbb{F}|=q=p^m\geq4$ where $p=\text{char}(\mathbb{F})$, I'm wondering is there a characterization of the kernel of the map $f:H_3(\text{GL}_3(\mathbb{F}))\...
XYC's user avatar
  • 389
12 votes
3 answers
503 views

Small simplicial set models for BG

Let $F$ be a finite group. Is there a model for $BF$ as a simplicial set such that the number of nondegenerate $n$-simplices grows at most polynomially? For example the Bar construction has the ...
HenrikRüping's user avatar
2 votes
0 answers
73 views

Homology of linear groups over integral domains and their field of fractions

Let $A$ be a noetherian integral domain of finite Krull dimension with the field of fractions of $F$. Consider the natural injections $i_n:GL_n(A)\hookrightarrow GL_{n+1}(A)$, $j_n:GL_n(F)\...
user127776's user avatar
  • 5,821
6 votes
0 answers
125 views

Localizations of group algebras of free groups

$\newcommand{\QQ}{\Bbb Q}$ Let $G$ be a free group on the symbols $x_1, \dots, x_n$, with $\QQ[G]$ its rational group algebra. Write $\varepsilon: \QQ[G] \to \QQ$ for the augmentation, and for $\...
Tyler Lawson's user avatar
13 votes
0 answers
256 views

Group homology $\mathrm{SL}_2$ acting on $\mathrm{Sym}^g$

Let $k$ be a field. We write $\mathrm{Sym}^g(k^2)$ for the $g$-th symmetric power of the (a?) standard representation of $\mathrm{GL}_2(k)$ ($g\geq 0$ an integer). Here I consider $\mathrm{Sym}^g(k^2)$...
tkr's user avatar
  • 408
7 votes
1 answer
373 views

Homology of symplectic groups in the unstable range

Let $Sp(2n,{\mathbb R})$ be the symplectic group and $H_3(Sp(2n,{\mathbb R});{\mathbb Z})$ its 3rd group homology (i.e., for the group with the discrete topology). It is known that $$H_3(Sp(2n,{\...
ThiKu's user avatar
  • 10.2k
3 votes
1 answer
260 views

Homology of solvable Lie groups made discrete

In what follows "homology" will mean group homology, i.e. $H_*(BG^\delta;{\mathbf R})$ for the group $G$ with the discrete topology. It is well-known how to compute the homology of abelian groups, ...
ThiKu's user avatar
  • 10.2k
4 votes
0 answers
175 views

Is the Tensor/Exterior square $G\otimes G$ or $G\wedge G$ of infinite p-group also a p-group?

Let $G$ be an infinite countable p-group. Is it true that $G\otimes G$ or $G\wedge G$ are also p-groups? (where G acts on itself by conjugation). For simplicity, you can consider that $G=[G,G]$, and ...
Alin Galatan's user avatar
4 votes
0 answers
84 views

Methods for showing homology of a subgroup survives to the larger group

Suppose we have an inclusion of groups $G_1<G_2$. I am curious about what methods there are out there for analyzing the map $H_k(G_1;\mathbb Q)\to H_k(G_2;\mathbb Q)$. In particular, what are tools ...
Jim Conant's user avatar
  • 4,838
1 vote
1 answer
455 views

Cup-products and Transgression maps.

This question is related to Lyndon-Hochschild-Serre spectral sequence and cup products. I have the followin result by J.S Milne in his book Arithmetic duality theorems pg 105. Let $$0 \rightarrow C \...
Chris Birkbeck's user avatar
1 vote
0 answers
292 views

Explicit transgression maps for Group homology in LHS

This question is related to another question of mine (Here: Lyndon-Hochschild-Serre spectral sequence and cup products). I'm trying to figure out if some diagram commutes an one of the maps involved ...
Chris Birkbeck's user avatar
2 votes
1 answer
1k views

Transgression maps in group cohomology and group homology / duality of spectral sequences

I am interested in whether the transgression maps for group cohomology and group homology are related via a version of the universal coefficient theorem. Let $G$ be a group, $H$ a normal subgroup of $...
Moshe Adrian's user avatar
  • 1,000