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We generalize results from topological robotics on the topological complexity (TC) of aspherical spaces to sectional categories of fibrations inducing subgroup inclusions on the level of fundamental groups. In doing so, we establish new lower bounds on sequential TCs of aspherical spaces as well as the parametrized TC of epimorphisms. Moreover, we generalize the Costa–Farber canonical class for TC to classes for sequential TCs and explore their properties. We combine them with the results on sequential TCs of aspherical spaces to obtain results on spaces that are not necessarily aspherical. [ABSTRACT FROM AUTHOR]

We introduce a criterion on the presentation of finitely presented pro-p groups which allows us to compute their cohomology groups and infer non-analytic quotients, with cohomological dimension strictly larger than two, from (non-free) mild groups. We interpret these groups as Galois groups over p-rational fields with prescribed ramification and splitting. [ABSTRACT FROM AUTHOR]

In this paper we study the simplicial complex induced by the poset of Brauer pairs ordered by inclusion for the family of finite reductive groups. In the defining characteristic case the homotopy type of this simplicial complex coincides with that of the Tits building thanks to a well-known result of Quillen. On the other hand, in the non-defining characteristic case, we show that the simplicial complex of Brauer pairs is homotopy equivalent to a simplicial complex determined by generalised Harish-Chandra theory. This extends earlier results of the author on the Brown complex and makes use of the theory of connected subpairs and twisted block induction developed by Cabanes and Enguehard. [ABSTRACT FROM AUTHOR]

Inspired by work of Szymik and Wahl on the homology of Higman–Thompson groups, we establish a general connection between ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces, based on the construction of small permutative categories of compact open bisections. This allows us to analyse homological invariants of topological full groups in terms of homology for ample groupoids.

Given a group G and a partial factor set σ of G, we introduce the twisted partial group algebra κ par σ G , which governs the partial projective σ -representations of G into algebras over a field κ. Using the relation between partial projective representations and twisted partial actions we endow κ par σ G with the structure of a crossed product by a twisted partial action of G on a commutative subalgebra of κ par σ G. Then, we use twisted partial group algebras to obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the crossed product A ∗ Θ G , involving the Hochschild homology of A and the partial homology of G, where Θ is a unital twisted partial action of G on a κ -algebra A with a κ -based twist. An analogous third quadrant cohomological spectral sequence is also obtained. [ABSTRACT FROM AUTHOR]

Let F be a non-cyclic free group of rank n. Consider the quotient F / [ F ″ , F ] , the free centre-by-metabelian group of rank n. In 1973, C. K. Gupta proved by purely group theoretic means that it contains an elementary abelian 2-group of rank ( 4 n) in its centre for n ≥ 4 , and exhibited an explicit generating set for this torsion subgroup. In this paper, using homological methods, we provide an alternative explicit generating set for it, and identify this torsion subgroup as the isolator of an explicitly given subgroup in the quotient F ″ / [ F ″ , F ′ ] . [ABSTRACT FROM AUTHOR]

Let (S , ω) be a closed connected oriented surface whose genus l is at least two equipped with a symplectic form. Then we show the vanishing of the cup product of the fluxes of commuting symplectomorphisms. This result may be regarded as an obstruction for commuting symplectomorphisms. In particular, the image of an abelian subgroup of Symp 0 c (S , ω) under the flux homomorphism is isotropic with respect to the natural intersection form on H 1 (S ; R) . The key to the proof is a refinement of the non-extendability result, previously given by the first-named and second-named authors, for Py's Calabi quasimorphism μ P on Ham c (S , ω) . [ABSTRACT FROM AUTHOR]

We investigate the minimal genus problem for the second homology of a right angled Artin group (RAAG). Firstly, we present a lower bound for the minimal genus of a second homology class, equal to half the rank of the corresponding cap product matrix. We show that for complete graphs, trees, and complete bipartite graphs, this bound is an equality, and furthermore in these cases the minimal genus can always be realised by a disjoint union of tori. Additionally, we give a full characterisation of classes that are representable by a single torus. However, the minimal genus of a second homology class of a RAAG is not always realised by a disjoint union of tori as an example we construct in the pentagon shows. [ABSTRACT FROM AUTHOR]

The aim of the paper is to give a full classification of factorizations of groups in terms of descent cohomology (pointed) sets introduced in Mesablishvili (Trans. A. Razmadze Math. Inst. 173(2), 137–155, 2019). We show that descent cohomology includes Serre's non-abelian group cohomology as a special case. This enables us to generalize Serre's theory further to include monoids. [ABSTRACT FROM AUTHOR]

We give a sharp bound on the number of triangles in a graph with fixed number of edges. We also characterize graphs that achieve the maximum number of triangles. Using the upper bound on number of triangles, we give a sharp bound for the size of the Schur multiplier of special p-group of all ranks. We also improve existing bounds for the size of Schur multiplier of p-groups (p ≠ 2 , 3) of class c ≥ 3 , and for p-groups of coclass r. [ABSTRACT FROM AUTHOR]

We investigate fixed-point properties of automorphisms of groups similar to Richard Thompson’s group F . Revisiting work of Gonçalves and Kochloukova, we deduce a cohomological criterion to detect infinite fixed-point sets in the abelianization, implying the so-called property R ∞ . Using the Bieri–Neumann–Strebel Σ -invariant and drawing from works of Gonçalves–Sankaran–Strebel and Zaremsky, we show that our tool applies to many F -like groups, including Stein’s group F 2 , 3 , cleary’s irrational-slope group F τ , the Lodha–Moore groups, and the braided version of F . [ABSTRACT FROM AUTHOR]

Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks of class groups of certain nonabelian extensions of $\mathbb {Q}$ and to give a new proof of the vanishing of Massey triple products in Galois cohomology.

The Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957. In 2011, a strengthened version of the conjecture was proved independently by Joel Friedman and by Igor Mineyev. In this paper we show that the strengthened Hanna Neumann conjecture holds not only in free groups but also in non-solvable surface groups. In addition, we show that a retract in a free group and in a surface group is inert. This implies the Dicks–Ventura inertia conjecture for free and surface groups. [ABSTRACT FROM AUTHOR]

For every one-relator monoid M = ⟨ A ∣ u = v ⟩ with u , v ∈ A ∗ we construct a contractible M-CW complex and use it to build a projective resolution of the trivial module which is finitely generated in all dimensions. This proves that all one-relator monoids are of type FP ∞ , answering positively a problem posed by Kobayashi in 2000. We also apply our results to classify the one-relator monoids of cohomological dimension at most 2, and to describe the relation module, in the sense of Ivanov, of a torsion-free one-relator monoid presentation as an explicitly given principal left ideal of the monoid ring. In addition, we prove the topological analogues of these results by showing that all one-relator monoids satisfy the topological finiteness property F ∞ , and classifying the one-relator monoids with geometric dimension at most 2. These results give a natural monoid analogue of Lyndon’s Identity Theorem for one-relator groups. [ABSTRACT FROM AUTHOR]

We finish off the classification of the endo-trivial modules of finite groups with Sylow 2-subgroups isomorphic to a semi-dihedral 2-group started by Carlson, Mazza and Thévenaz in their article Endotrivial modules over groups with quaternion or semi-dihedral Sylow 2-subgroup published in 2013. [ABSTRACT FROM AUTHOR]

We study the Bieri–Neumann–Strebel–Renz invariants and we prove the following criterion: for groups H and K of type F ⁢ P n {FP_{n}} such that [ H , H ] ⊆ K ⊆ H {[H,H]\subseteq K\subseteq H} and a character χ : K → ℝ {\chi:K\to\mathbb{R}} with χ ⁢ ([ H , H ]) = 0 {\chi([H,H])=0} , we have [ χ ] ∈ Σ n ⁢ (K , ℤ) {[\chi]\in\Sigma^{n}(K,\mathbb{Z})} if and only if [ μ ] ∈ Σ n ⁢ (H , ℤ) {[\mu]\in\Sigma^{n}(H,\mathbb{Z})} for every character μ : H → ℝ {\mu:H\to\mathbb{R}} that extends χ. The same holds for the homotopical invariants Σ n ⁢ (-) {\Sigma^{n}(-)} when K and H are groups of type F n {F_{n}}. We use these criteria to complete the description of the Σ-invariants of the Bieri–Stallings groups G m {G_{m}} , and more generally to describe the Σ-invariants of the Bestvina–Brady groups. We also show that the "only if" direction of the above criterion holds if we assume only that K is a subnormal subgroup of H, where both groups are of type F ⁢ P n {FP_{n}}. We apply this last result to wreath products. [ABSTRACT FROM AUTHOR]

We say a group G satisfies properties (M) and (NM) if every nontrivial finite subgroup of G is contained in a unique maximal finite subgroup, and every nontrivial finite maximal subgroup is self-normalizing. We prove that the Bredon cohomological dimension and the virtual cohomological dimension coincide for groups that admit a cocompact model for EG and satisfy properties (M) and (NM). Among the examples of groups satisfying these hypothesis are cocompact and arithmetic Fuchsian groups, one-relator groups, the Hilbert modular group, and 3-manifold groups. [ABSTRACT FROM AUTHOR]

Let F be a free group of rank d ≥ 2. Consider the quotient F / [ γ 4 (F ′) , F ] F ″ ′ , the free centre-by-(nilpotent of class 3)-by-abelian group within the variety of all soluble groups of derived length 3. A peculiar feature of this group is the occurence of torsion in its centre. Our main result is a complete description of an elementary abelian 2-group of rank ( d 4 ) in F / [ γ 4 (F ′) , F ] F ″ ′ in terms of generators of F for d ≥ 4. Our approach makes essential use of homological methods. [ABSTRACT FROM AUTHOR]

Presentations of groups by rewriting systems (that is, by monoid presentations), have been fruitfully studied by encoding the rewriting system in a 2-complex—the Squier complex—whose fundamental groupoid then describes the derivation of consequences of the rewrite rules. We describe a reduced form of the Squier complex, investigate the structure of its fundamental groupoid, and show that key properties of the presentation are still encoded in the reduced form. [ABSTRACT FROM AUTHOR]

In this short survey article, we aim to provide an up to date information on the progress made towards Schurs exponent property and related conjectures. We also mention the connection between Schurs exponent property and Noether's Rationality problem. [ABSTRACT FROM AUTHOR]

We build on the constructions in Brady (J Lond Math Soc 60(2):461–480, 1999) and Lodha (A hyperbolic group with a finitely presented subgroup that is not of type FP3, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, pp 67–81, 2017) to give infinite families of hyperbolic groups, each having a finitely presented subgroup that is not of type F 3 . By calculating the Euler characteristic of the hyperbolic groups constructed, we prove that infinitely many of them are pairwise non isomorphic. We further show that the first of these constructions cannot be generalised to dimensions higher than 3. [ABSTRACT FROM AUTHOR]

The Σ-invariants of Bieri–Neumann–Strebel and Bieri–Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Σ-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The "zeroth stage" of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the "nth stage" for any n. [ABSTRACT FROM AUTHOR]

Let G 2 be a group which acts trivially on an abelian group G 1 . According to the Schreier's Theorem, each 2-cocycle ε ∈ Z 2 (G 2 , G 1) determines a group G ε which is a central extension of G 1 by G 2 , and we will denote this group by G 1 × ε G 2 and call it the perturbed direct product of G 1 by G 2 under ε . The aim of this paper is to study properties of the perturbed direct products. For two distinct 2-cocycles ε 1 and ε 2 , we find necessary and sufficient conditions for G 1 × ε 1 G 2 to be isomorphic to G 1 × ε 2 G 2 . Furthermore, we obtain some results about decompositions for a given perturbed direct product G 1 × ε G 2 when G 1 or G 2 is a nontrivial direct product. [ABSTRACT FROM AUTHOR]

We establish Green equivalences for all Mackey 2-functors, without assuming Krull–Schmidt. By running through the examples of Mackey 2-functors, we recover all variants of the Green equivalence and Green correspondence known in representation theory and obtain new ones in several other contexts. Such applications include equivariant stable homotopy theory in topology and equivariant sheaves in geometry. [ABSTRACT FROM AUTHOR]

For any given subgroup H of a finite group G, the Quillen poset ${\mathcal {A}}_p(G)$ of nontrivial elementary abelian p-subgroups is obtained from ${\mathcal {A}}_p(H)$ by attaching elements via their centralisers in H. We exploit this idea to study Quillen’s conjecture, which asserts that if ${\mathcal {A}}_p(G)$ is contractible then G has a nontrivial normal p-subgroup. We prove that the original conjecture is equivalent to the ${{\mathbb {Z}}}$-acyclic version of the conjecture (obtained by replacing ‘contractible’ by ‘${{\mathbb {Z}}}$-acyclic’). We also work with the ${\mathbb {Q}}$-acyclic (strong) version of the conjecture, reducing its study to extensions of direct products of simple groups of p-rank at least $2$. This allows us to extend results of Aschbacher and Smith and to establish the strong conjecture for groups of p-rank at most $4$.

We study the fundamental group of the p‐subgroup complex of a finite group G. We show first that π1(A3(A10)) is not a free group (here A10 is the alternating group on ten letters). This is the first concrete example in the literature of a p‐subgroup complex with non‐free fundamental group. We prove that, modulo a well‐known conjecture of Aschbacher, π1(Ap(G))=π1(Ap(SG))∗F, where F is a free group and π1(Ap(SG)) is free if SG is not almost simple. Here SG=Ω1(G)/Op′(Ω1(G)). This result essentially reduces the study of the fundamental group of p‐subgroup complexes to the almost simple case. We also exhibit various families of almost simple groups whose p‐subgroup complexes have free fundamental group. [ABSTRACT FROM AUTHOR]

New constructions in group homology allow us to manufacture high-dimensional manifolds with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of simplicial volumes of orientable closed connected manifolds is dense in R ≥ 0 . In dimension 4 we prove that every non-negative rational number is the simplicial volume of some orientable closed connected 4-manifold. Our group theoretic results relate stable commutator length to the l 1 -semi-norm of certain singular homology classes in degree 2. The output of these results is translated into manifold constructions using cross-products and Thom realisation. [ABSTRACT FROM AUTHOR]

For any given subgroup H of a finite group G, the Quillen poset ${\mathcal {A}}_p(G)$ of nontrivial elementary abelian p-subgroups is obtained from ${\mathcal {A}}_p(H)$ by attaching elements via their centralisers in H. We exploit this idea to study Quillen's conjecture, which asserts that if ${\mathcal {A}}_p(G)$ is contractible then G has a nontrivial normal p-subgroup. We prove that the original conjecture is equivalent to the ${{\mathbb {Z}}}$ -acyclic version of the conjecture (obtained by replacing 'contractible' by ' ${{\mathbb {Z}}}$ -acyclic'). We also work with the ${\mathbb {Q}}$ -acyclic (strong) version of the conjecture, reducing its study to extensions of direct products of simple groups of p-rank at least $2$. This allows us to extend results of Aschbacher and Smith and to establish the strong conjecture for groups of p-rank at most $4$. [ABSTRACT FROM AUTHOR]

We introduce the notions of overcommutation and overcommutation length in groups, and show that these concepts are closely related to representations of the fundamental groups of 3‐manifolds and their Heegaard genus. We give many examples including translations in the affine group of the line and provide upper bounds for the overcommutation length in SL2, related to the Steinberg relation. [ABSTRACT FROM AUTHOR]

We calculate the Bieri–Neumann–Strebel–Renz invariant Σ1(G) for finitely presented residually free groups G and show that its complement in the character sphere S(G) is a finite union of finite intersections of closed sub-spheres in S(G). Furthermore, we find some restrictions on the higher-dimensional homological invariants Σn(G, ℤ) and show for the discrete points Σ2(G)dis, Σ2(G, ℤ)dis and Σ2(G, ℚ)dis in Σ2(G), Σ2(G, ℤ) and Σ2(G, ℚ) that we have the equality Σ2(G)dis = Σ2(G, ℤ)dis = Σ2(G, ℚ)dis. [ABSTRACT FROM AUTHOR]

In this paper, we deal with stable homology computations with twisted coefficients for mapping class groups of surfaces and of 3-manifolds, automorphism groups of free groups with boundaries and automorphism groups of certain right-angled Artin groups. On the one hand, the computations are led using semidirect product structures arising naturally from these groups. On the other hand, we compute the stable homology with twisted coefficients by FI-modules. This notably uses a decomposition result of the stable homology with twisted coefficients for pre-braided monoidal categories proved in this paper. Communicated by Jason P. Bell [ABSTRACT FROM AUTHOR]

We study homological approximations of the profinite completion of a limit group (see Thm. A) and obtain the analogous of Bridson and Howie's Theorem for the profinite completion of a non-abelian limit group (see Thm. B). [ABSTRACT FROM AUTHOR]

We investigate the universal cover of a topological group that is not necessarily connected. Its existence as a topological group is governed by a Taylor cocycle, an obstruction in 3-cohomology. Alternatively, it always exists as a topological 2-group. The splitness of this 2-group is also governed by an obstruction in 3-cohomology, a Sinh cocycle. We give explicit formulas for both obstructions and show that they are equal. [ABSTRACT FROM AUTHOR]

Suppose S is a closed orientable surface and S ~ is a finite sheeted regular cover of S. When studying mapping class groups, the following question arose: do the lifts of simple curves from S generate H 1 (S ~ , Z) ? A family of examples is given for which the answer is "no". [ABSTRACT FROM AUTHOR]

We prove that Thompson's group V is acyclic, answering a 1992 question of Brown in the positive. More generally, we identify the homology of the Higman–Thompson groups V n , r with the homology of the zeroth component of the infinite loop space of the mod n - 1 Moore spectrum. As V = V 2 , 1 , we can deduce that this group is acyclic. Our proof involves establishing homological stability with respect to r, as well as a computation of the algebraic K-theory of the category of finitely generated free Cantor algebras of any type n. [ABSTRACT FROM AUTHOR]

Does a space enjoying good finiteness properties admit an algebraic model with commensurable finiteness properties? In this note, we provide a rational homotopy obstruction for this to happen. As an application, we show that the maximal metabelian quotient of a very large, finitely generated group is not finitely presented. Using the theory of 1‐minimal models, we also show that a finitely generated group π admits a connected 1‐model with finite‐dimensional degree 1 piece if and only if the Malcev Lie algebra m(π) is the lower central series completion of a finitely presented Lie algebra. [ABSTRACT FROM AUTHOR]

We consider the group X(G) obtained from G*G by forcing each element g in the first free factor to commute with the copy of g in the second free factor. Deceptively complicated finitely presented groups arise from this construction: X(G) is finitely presented if and only if G is finitely presented, but if F is a non‐abelian free group of finite rank then X(F) has a subgroup of finite index whose third homology is not finitely generated. [ABSTRACT FROM AUTHOR]

We show that every countable group embeds in a group of type F P 2 . [ABSTRACT FROM AUTHOR]

Abstract: We construct uncountably many discrete groups of type F P; in particular we construct groups of type F P that do not embed in any finitely presented group. We compute the ordinary, ℓ 2, and compactly supported cohomology of these groups. For each n ⩾ 4 we construct a closed aspherical n‐manifold that admits an uncountable family of acyclic regular coverings with non‐isomorphic covering groups. [ABSTRACT FROM AUTHOR]

We study the homotopy properties of the posets of p -subgroups S p ( G ) and A p ( G ) of a finite group G , viewed as finite topological spaces. We answer a question raised by R.E. Stong in 1984 about the relationship between the contractibility of the finite space A p ( G ) and that of S p ( G ) negatively, and describe the contractibility of A p ( G ) in terms of algebraic properties of the group G . [ABSTRACT FROM AUTHOR]

We consider smooth, complex quasiprojective varieties $U$ that admit a compactification with a boundary, which is an arrangement of smooth algebraic hypersurfaces. If the hypersurfaces intersect locally like hyperplanes, and the relative interiors of the hypersurfaces are Stein manifolds, we prove that the cohomology of certain local systems on $U$ vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements are both duality and abelian duality spaces.

Let G be an infinite discrete group. A classifying space for proper actions of G is a proper G‐CW complex X such that the fixed point sets XH are contractible for all finite subgroups H of G. In this paper we consider the stable analogue of the classifying space for proper actions in the category of proper G‐spectra and study its finiteness properties. We investigate when G admits a stable classifying space for proper actions that is finite or of finite type and relate these conditions to the compactness of the sphere spectrum in the homotopy category of proper G‐spectra and to classical finiteness properties of the Weyl groups of finite subgroups of G. Finally, if the group G is virtually torsion‐free we also show that the smallest possible dimension of a stable classifying space for proper actions coincides with the virtual cohomological dimension of G, thus providing the first geometric interpretation of the virtual cohomological dimension of a group. [ABSTRACT FROM AUTHOR]