# Tagged Questions

**8**

votes

**1**answer

105 views

### Torsion-free group that is not of type F but is virtually of type F

Recall that a group $G$ is of type F if there exists a compact $K(G,1)$.
There are many examples of groups which are not of type F but which are virtually of type F, that is, they have finite-index ...

**1**

vote

**0**answers

86 views

### Reference on calculation of 2nd cohomology group

Let $G$ be a finitely generated, infinite, countable discrete nonamenable group with zero first Betti number, I.e., $H^1(G, \ell^2(G))=0$, e.g., $G=F_2\times F_2$, the product of free groups of two ...

**1**

vote

**1**answer

105 views

### Successive Schur covers

Let $G_0$ be a finite group and $G_j$ a Schur cover of $G_{j-1}$ for $j=1,2,3\ldots$. Is $G_2$ equal to $G_1$? If not, will the sequence stop after finite steps in general?

**2**

votes

**1**answer

220 views

### vanishing higher cohomology group for property T group?

Given a countable discrete group $G$ with Kazhdan's property (T), consider $\mathbb{C}G$ or $l^2(G)$ as a left $G$-module, then we can consider the group cohomology,
Is it known that $H^n(G, ...

**7**

votes

**1**answer

249 views

### Is there a finitely presented group with infinite homology over $\mathbb{Q}$?

Suppose $G$ is a discrete group given by finitely many generators with finitely many relations. Can the homology groups $H_i(G, \mathbb{Q})$, or equivalently $H_i(BG, \mathbb{Q})$ (topological ...

**0**

votes

**1**answer

115 views

### Known computations of certain 2-cohomology groups?

I wanted to know if there are any computations of cohomology groups $H^n(\Gamma,A^{(\Gamma)})$ in the literature for certain $n\in\mathbb{N}$, Abelian groups $A$, and infinite groups $\Gamma$.
Here ...

**9**

votes

**0**answers

218 views

### Non split extension isomorphic (as a group) to a split extension

$\def\Z{\mathbb{Z}}$
Let $A$ be a finite abelian group and $G$ a finite group acting on $A$.
Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is ...

**4**

votes

**1**answer

214 views

### Group extensions isomorphic as groups

Let $G$ be a group and $A$ a $G$-module. It well know that there is a group isomorphism between the second cohomologoy group $H^2(G,A)$ and the abelian group $OpExt(G,A)$ of classes of extension ...

**3**

votes

**0**answers

90 views

### Groups such that all finite-dim representations are finitely presented

Let $G$ be an infinite group. What sorts of finiteness properties can I put on $G$ to ensure the following holds for all $M$?
Let $M$ be a finite-dimensional vector space over $\mathbb{Q}$ upon ...

**7**

votes

**2**answers

213 views

### 3-cocycle representatives for the dihedral group $D_{2n}$?

I am looking for a reference for a complete list of 3-cocycle representatives for $H^3(D_{2n},\mathbb{C}^\times)$, where
$$
D_{2n}=\langle a, b\mid a^2=b^2=(ab)^n=e\rangle
$$
is the dihedral group of ...

**11**

votes

**1**answer

241 views

### Group cochains invariant under the action of the symmetric group

Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups
$$
C^k = \{f: G^k \to A\}
$$
and the coboundary map
$$
\delta : C^k \to C^{k+1}
$$
$$
(\delta f)(g_1, \ldots, g_{k+1}) ...

**9**

votes

**1**answer

252 views

### Calculations of nonabelian group cohomology of R^n

I am looking at $H^1(\mathbb{R}^n,G)$ where $G$ is a finite 2-group. I'm wondering if such things have been calculated. I'm afraid I can't say I know anything here, past the result that this ...

**11**

votes

**1**answer

362 views

### Do there exist non-isomorphic groups with the same cohomology?

For any group $G$, cohomology can be viewed as a functor
$$
H^\ast(G,-): G{\sf\text{-}mod}\to {\sf GrAbGrp},
$$
where $G{\sf\text{-}mod}$ denotes the category of (left) $\mathbb{Z}[G]$-modules and ...

**9**

votes

**1**answer

428 views

### A map from the coinvariants of the dual to the dual of the invariants for a G-module

Suppose $G$ is a group and $X$ is a $\mathbb{Z}[G]$-module. Recall that the augmentation ideal $I \subset \mathbb{Z}[G]$ is generated by elements of the form $g - 1$ for $g \in G$, the coinvariants ...

**4**

votes

**1**answer

190 views

### Interesting families of groups as group extensions

Let me start this question with an example that hopefully makes clear what I am looking for:
A discrete subgroup $G$ of the group of euclidean isometries of $\mathbb{R}^d$ is called a ...

**7**

votes

**2**answers

330 views

### A question on some computation of group cohomologies

Let $G=H\times J$, where $H\cong J\cong C_2$ (cyclic group of order 2). Let $M \cong \mathbb{Z}$ be a $G$-module via "trivial $H$-action and negation $J$-action". My question is "What are the group ...

**2**

votes

**1**answer

154 views

### A sort of “group-ring” construction on coefficient systems in group homology (+ special case involving GL(n,Z))

Let $G$ be a discrete group and $M$ be an $RG$-module for some ring $R$ (I'm happy to assume that $R = \mathbb{Q}$). Define $R[M]$ to be the set of $R$-linear combinations of formal symbols of the ...

**2**

votes

**1**answer

94 views

### Resolution of coefficient system in group homology

Let $G$ be a discrete group and let $M$ be a $G$-module. Assume that I have a resolution
$$\cdots \rightarrow M_1 \rightarrow M_0 \rightarrow M \rightarrow 0$$
of $M$ by $G$-modules (with no further ...

**4**

votes

**3**answers

444 views

### Reference for Ring Structure on Group Cohomology

As a graded $\mathbb{Z}$-module, the structure of the group cohomology $H^{*}(\mathbb{Z}/n\mathbb{Z};\mathbb{Z})$ is extremely well-known. Yet, I am having difficulty finding a reference concerning ...

**9**

votes

**2**answers

300 views

### Cohomological dimension of knit products

Let $G$ be a group with complementary subgroups $A$ and $B$ (meaning $A\cap B=1$ and $AB=G$).
If $A$ and $B$ are both normal in $G$, then $G\cong A\times B$ is a direct product. If $A$ is normal, ...

**3**

votes

**1**answer

145 views

### A group 3-cocycle, trivial on a pair of generating subgroups?

I'm looking for an example of the following situation:
A group $G$ generated by finite subgroups $H$ and $K$,
a non-trivial 3-cocycle $\omega \in H^3(G, \mathbb{k}^\times)$
such that
the ...

**5**

votes

**1**answer

429 views

### Can group cohomology be interpreted as an obstruction to lifts?

The standard way to view the first and second group cohomologies is this:
The Standard Story
Let $G$ be a group, and let $M$ be a commutative group with a $G$-action. Then the first cohomology has ...

**3**

votes

**1**answer

250 views

### Cohomological dimension of groups & number of generators

I have a torsion-free non-abelian nilpotent group $\Gamma$ of cohomological dimension $n$. Is it possible to say anything about the number of generators of $\Gamma$ in a minimal presentation?
Can I ...

**3**

votes

**2**answers

229 views

### Homology groups of divisible and powered (nilpotent) groups

(1) Suppose $\pi$ is a set of primes and $G$ is a $\pi$-divisible nilpotent group, i.e., for any $g \in G$ and $p \in \pi$, there exists $x \in G$ such that $x^p = g$. Is it necessary that all the ...

**2**

votes

**3**answers

368 views

### Computing the cardinality of cohomology groups

I hope this question is not unreasonably broad. It is about calculating or at least bounding the cardinality of cohomology groups in case they are finite.
Let us assume we are given a group $G$ and a ...

**7**

votes

**1**answer

176 views

### Nonhyperbolic groups that contain no free abelian groups or Baumslag-Solitar groups

I've heard it conjectured that a finitely presentable group $G$ is hyperbolic if it satisfies the following two conditions.
$G$ contains no subgroup isomorphic to a Baumslag-Solitar group $BS(n,m)$ ...

**13**

votes

**3**answers

444 views

### Does this subgroup of “even braids” have a name?

The full braid group on $n$ strands $B_n$ admits a surjective homomorphism $p\colon\thinspace B_n\to \Sigma_n$ onto the symmetric group on $n$ letters, which takes a braid to the induced permutation ...

**2**

votes

**1**answer

214 views

### Naturality of the transfer in group cohomology

Let $G$ be a (discrete) group and $H\le G$ a subgroup of finite index. Then there is a transfer map
$$tr\colon\thinspace H^\ast(H;M)\to H^\ast(G;M)
$$
in group cohomology, where $M$ is any $G$-module ...

**3**

votes

**1**answer

107 views

### Dimension of the cohomology ring of an extension of groups

Given an extension $1 \to N \to P \to Q \to 1$ of p-groups. Is it true that
$$\dim H^\ast(P,\mathbb{F}_p) = \dim \text{im}(res^P_N) + \dim \text{im}(inf^P_Q)$$
where $\dim$ denotes the Krull ...

**1**

vote

**0**answers

97 views

### Extensions of orthogonal groups of torsion quadratic forms.

Hi. This is related to a question I asked earlier. The setup is:
Let $L$ be an $n$-dimensional lattice with an integer valued quadratic form $q$. Fix a basis $e_i$ for $L$ and let $K_{ij} = ...

**18**

votes

**2**answers

845 views

### When does a homomorphism factor through a free group?

Let $f\colon\thinspace G\to H$ be a surjective homomorphism of finitely generated groups. Are there any methods to decide whether $f$ factors through a free group? That is, does there exist a free ...

**2**

votes

**1**answer

150 views

### Calculating $H^n(G, \mathbb{Z}G)$ as co-homology with compact support of a proper co-compact $G$-CW-complex $X$

This question was originally posted to Math.StackExchange, but having got no response there, I'm reposting it here. I apologise if it is too elementary for this site.
(Original post: ...

**6**

votes

**2**answers

468 views

### cohomological dimension of a group acting on a product

I recently came across an interesting result of Kobayashi [Corollary 5.5], a special case of which is the following: Suppose $\Gamma$ is a discrete torsion free subgroup of $SL_n(\mathbb{R})$ which ...

**4**

votes

**1**answer

140 views

### For which rings R is SL_n(R) a virtual duality group

A famous theorem of Borel and Serre says that if $R$ is the ring of integers in an algebraic number field, then $\text{SL}_n(R)$ satisfies virtual Bieri-Eckmann duality. In other words, there exists ...

**12**

votes

**1**answer

827 views

### Interpretation of universal coefficients theorem for group cohomology

Suppose $G$ and $A$ are abelian groups (I'm setting $G$ abelian to keep the discussion simple, though there are analogues for non-abelian $G$) with $G$ acting trivially on $A$. By the universal ...

**3**

votes

**2**answers

354 views

### Cohomological dimension of finitely presented group

I have a group of cohomological dimension 2 generated by two elements. Is it possible to deduce that the group is commutative or, more generally, does $\mathrm{cd}\ G=2$ imply anything about the ...

**0**

votes

**0**answers

85 views

### Isomorphisms of group extensions arising from antisymmetric forms

Let $V,W$ be topological vector spaces and fix continuous antisymmetric bilinear forms $\omega_1:V\times V\to \mathbb{R}$, $\omega_2:W\times W\to\mathbb{R}$. Since $\omega_1$ is a 2-cocycle (in fact ...

**1**

vote

**1**answer

348 views

### Finiteness theorems for profinite groups

Let $G$ be a profinite group which fits in the following short exact sequence:
$$
1\rightarrow N\rightarrow G \rightarrow K\rightarrow 1
$$
Assume that $N$ is a pro-$p$ group and that $K$ is ...

**17**

votes

**2**answers

743 views

### Proofs of the Stallings-Swan theorem

It is a well-known and deep${}^\ast$ theorem that if a group $G$ has cohomological dimension one then it must be free. This was proved in the late 60's by Stallings (for finitely generated groups) and ...

**4**

votes

**1**answer

282 views

### Transgressions commute with the Steenrod operations on the base and fiber in a central group extension?

The following sentence is quoted from the paper ON THE COHOMOLOGY OF SPLIT EXTENSIONS by D. J. BENSON AND M. FESHBACH:
In general, the differentials in the Lyndon-Hochschild-Serre spectral sequence
...

**3**

votes

**1**answer

768 views

### First group homology with general coefficients

When $G$ acts trivially on $M$, the first homology group is just the abelianisation of $G$ tensored with $M$, i.e. $H_1(G;M)=(G/[G,G])\otimes_\mathbb Z M$.
Is there any similar statement when $G$ ...

**1**

vote

**0**answers

138 views

### Twisted homology of free products

Let $G_1$ and $G_2$ be groups and let $M$ be a vector space equipped with actions of $G_1$ and $G_2$. The free product $G_1 \ast G_2$ thus acts on $M$. How can one compute the twisted group homology ...

**10**

votes

**1**answer

493 views

### Second homology group of free nilpotent p-group

Let $F_n$ be a free group on $n$ generators. Fix a prime $p$. Let $\gamma_k^p(F_n)$ be the mod $p$ lower central series, i.e. the inductively defined series
$$\gamma_0^p(F_n) = F_n \quad \text{and} ...

**9**

votes

**2**answers

578 views

### Cohomological dimension of a homomorphism

Let $G$ and $\Gamma$ be discrete groups, and let $\phi\colon\thinspace G\to \Gamma$ be a homomorphism.
Define its cohomological dimension $\operatorname{cd}\phi$ to be the least integer $d$ such that ...

**22**

votes

**3**answers

949 views

### Geometric Interpretation of the Lower Central Series for the Fundamental Group?

For any group G we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain
$G_0 > G_1 > ... > ...

**13**

votes

**6**answers

2k views

### Characterization of the transfer map in group theory

Let $i : H \to G$ be a subgroup of finite index. The transfer map is a special homomorphism $V(i) : G^\mathrm{ab} \to H^\mathrm{ab}$. The usual ad hoc definition uses a set of representatives of $H$ ...

**2**

votes

**0**answers

305 views

### Inseparable Galois Cohomology

First let me give a general form of my question, and then I'll give some motivation and a more specific version of it. Let $K/k$ be a Galois extension of fields with Galois group $G$, and let $X$ be ...

**3**

votes

**1**answer

363 views

### Hirsch length and cohomological dimension

It seems to be known that Hirsch length and cohomological dimension agree for (torsion-free, finitely generated) polycyclic groups.
If we drop the assumption "torsion-free", then cd is of course ...

**6**

votes

**1**answer

354 views

### Posets of cosets and contractibility

For this question let $G$ be a group, perhaps infinite, and let $H_i$ for $i\in I$ be a (finite) family of subgroups closed under taking intersections. I am interested in the coset poset ...

**16**

votes

**1**answer

944 views

### Grothendieck's question on the Brauer group for groups

Let $G$ be a group, and let $M(G)=H^2(G,\mathbb{C}^*)$ be the Schur multiplier of $G$. There is a group $Br(G)$ of complex projective representations of $G$ modulo those that can be lifted to linear ...