# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $$\delta : C^k \to C^{k+1}$$ $$(\delta f)(g_1, \ldots, g_{k+1}) ... 1answer 253 views ### Calculations of nonabelian group cohomology of R^n I am looking at H^1(\mathbb{R}^n,G) where G is a finite 2-group. I'm wondering if such things have been calculated. I'm afraid I can't say I know anything here, past the result that this ... 1answer 365 views ### Do there exist non-isomorphic groups with the same cohomology? For any group G, cohomology can be viewed as a functor$$ H^\ast(G,-): G{\sf\text{-}mod}\to {\sf GrAbGrp}, $$where G{\sf\text{-}mod} denotes the category of (left) \mathbb{Z}[G]-modules and ... 1answer 432 views ### A map from the coinvariants of the dual to the dual of the invariants for a G-module Suppose G is a group and X is a \mathbb{Z}[G]-module. Recall that the augmentation ideal I \subset \mathbb{Z}[G] is generated by elements of the form g - 1 for g \in G, the coinvariants ... 1answer 190 views ### Interesting families of groups as group extensions Let me start this question with an example that hopefully makes clear what I am looking for: A discrete subgroup G of the group of euclidean isometries of \mathbb{R}^d is called a ... 2answers 332 views ### A question on some computation of group cohomologies Let G=H\times J, where H\cong J\cong C_2 (cyclic group of order 2). Let M \cong \mathbb{Z} be a G-module via "trivial H-action and negation J-action". My question is "What are the group ... 1answer 155 views ### A sort of “group-ring” construction on coefficient systems in group homology (+ special case involving GL(n,Z)) Let G be a discrete group and M be an RG-module for some ring R (I'm happy to assume that R = \mathbb{Q}). Define R[M] to be the set of R-linear combinations of formal symbols of the ... 1answer 106 views ### Resolution of coefficient system in group homology Let G be a discrete group and let M be a G-module. Assume that I have a resolution$$\cdots \rightarrow M_1 \rightarrow M_0 \rightarrow M \rightarrow 0$$of M by G-modules (with no further ... 3answers 447 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 ... 2answers 302 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, ... 1answer 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 ... 1answer 438 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 ... 1answer 252 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 ... 2answers 231 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 ... 3answers 370 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 ... 1answer 177 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) ... 3answers 446 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 ... 1answer 215 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) $$in group cohomology, where M is any G-module ... 1answer 107 views ### Dimension of the cohomology ring of an extension of groups Given an extension 1 \to N \to P \to Q \to 1 of p-groups. Is it true that$$\dim H^\ast(P,\mathbb{F}_p) = \dim \text{im}(res^P_N) + \dim \text{im}(inf^P_Q)$$where \dim denotes the Krull ... 0answers 98 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} = ... 2answers 858 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 ... 1answer 155 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: ... 2answers 471 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 ... 1answer 144 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 ... 1answer 829 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 ... 2answers 357 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 ... 0answers 86 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 ... 1answer 351 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 $$Assume that N is a pro-p group and that K is ... 2answers 747 views ### Proofs of the Stallings-Swan theorem It is a well-known and deep{}^\ast theorem that if a group G has cohomological dimension one then it must be free. This was proved in the late 60's by Stallings (for finitely generated groups) and ... 1answer 286 views ### Transgressions commute with the Steenrod operations on the base and fiber in a central group extension? The following sentence is quoted from the paper ON THE COHOMOLOGY OF SPLIT EXTENSIONS by D. J. BENSON AND M. FESHBACH: In general, the differentials in the Lyndon-Hochschild-Serre spectral sequence ... 1answer 777 views ### First group homology with general coefficients When G acts trivially on M, the first homology group is just the abelianisation of G tensored with M, i.e. H_1(G;M)=(G/[G,G])\otimes_\mathbb Z M. Is there any similar statement when G ... 0answers 138 views ### Twisted homology of free products Let G_1 and G_2 be groups and let M be a vector space equipped with actions of G_1 and G_2. The free product G_1 \ast G_2 thus acts on M. How can one compute the twisted group homology ... 1answer 494 views ### Second homology group of free nilpotent p-group Let F_n be a free group on n generators. Fix a prime p. Let \gamma_k^p(F_n) be the mod p lower central series, i.e. the inductively defined series$$\gamma_0^p(F_n) = F_n \quad \text{and} ...
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 ...