On the Finiteness length of some soluble linear groups

Rego, Yuri Santos

doi:10.4153/s0008414x21000213

Cited by 3 publications

(1 citation statement)

References 58 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this form, coset complexes were introduced by Abels and Holz in [1] but they appear with different names in several branches of group theory: The main motivation of Abels and Holz was to study finiteness properties of groups. Recent work in this direction can be found in the work of Bux, Fluch, Marschler, Witzel and Zaremsky [11] and Santos‐Rego [39]. In [35], Meier, Meinert and VanWyk used these complexes to study the BNS invariants of right‐angled Artin groups.…”

Section: Coset Posets and Coset Complexesmentioning

confidence: 99%

Between buildings and free factor complexes: A Cohen–Macaulay complex for Out(RAAGs)

Brück

2022

Journal of London Math Soc

View full text Add to dashboard Cite

For every finite graph Γ$\Gamma$, we define a simplicial complex associated to the outer automorphism group of the right‐angled Artin group (RAAG) AnormalΓ$A_\Gamma$. These complexes are defined as coset complexes of parabolic subgroups of Out0(AnormalΓ)$\operatorname{Out}^0(A_\Gamma )$ and interpolate between Tits buildings and free factor complexes. We show that each of these complexes is homotopy Cohen–Macaulay and in particular homotopy equivalent to a wedge of d$d$‐spheres. The dimension d$d$ can be read off from the defining graph Γ$\Gamma$ and is determined by the rank of a certain Coxeter subgroup of Out0(AnormalΓ)$\operatorname{Out}^0(A_\Gamma )$. To show this, we refine the decomposition sequence for Out0(AnormalΓ)$\operatorname{Out}^0(A_\Gamma )$ established by Day–Wade, generalise a result of Brown concerning the behaviour of coset posets under short exact sequences and determine the homotopy type of free factor complexes associated to relative automorphism groups of free products.

show abstract

Section: Coset Posets and Coset Complexesmentioning

confidence: 99%

Between buildings and free factor complexes: A Cohen–Macaulay complex for Out(RAAGs)

Brück

2022

Journal of London Math Soc

View full text Add to dashboard Cite

show abstract

Twisted conjugacy in soluble arithmetic groups

Lins de Araujo,

Santos Rego

2024

Mathematische Nachrichten

View full text Add to dashboard Cite

Reidemeister numbers of group automorphisms encode the number of twisted conjugacy classes of groups and might yield information about self‐maps of spaces related to the given objects. Here, we address a question posed by Gonçalves and Wong in the mid‐2000s: we construct an infinite series of compact connected solvmanifolds (that are not nilmanifolds) of strictly increasing dimensions and all of whose self‐homotopy equivalences have vanishing Nielsen number. To this end, we establish a sufficient condition for a prominent (infinite) family of soluble linear groups to have the so‐called property . In particular, we generalize or complement earlier results due to Dekimpe, Gonçalves, Kochloukova, Nasybullov, Taback, Tertooy, Van den Bussche, and Wong, showing that many soluble ‐arithmetic groups have and suggesting a conjecture in this direction.

show abstract

The Σ-invariants of 𝑆-arithmetic subgroups of Borel groups

Schesler

2023

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Given a Chevalley group G \mathcal {G} of classical type and a Borel subgroup B ⊆ G \mathcal {B} \subseteq \mathcal {G} , we compute the Σ \Sigma -invariants of the S S -arithmetic groups B ( Z [ 1 / N ] ) \mathcal {B}(\mathbb {Z}[1/N]) , where N N is a product of large enough primes. To this end, we let B ( Z [ 1 / N ] ) \mathcal {B}(\mathbb {Z}[1/N]) act on a Euclidean building X X that is given by the product of Bruhat–Tits buildings X p X_p associated to G \mathcal {G} , where p p is a prime dividing N N . In the course of the proof we introduce necessary and sufficient conditions for convex functions on C A T ( 0 ) CAT(0) -spaces to be continuous. We apply these conditions to associate to each simplex at infinity τ ⊂ ∂ ∞ X \tau \subset \partial _\infty X its so-called parabolic building X τ X^{\tau } and to study it from a geometric point of view. Moreover, we introduce new techniques in combinatorial Morse theory, which enable us to take advantage of the concept of essential n n -connectivity rather than actual n n -connectivity. Most of our building theoretic results are proven in the general framework of spherical and Euclidean buildings. For example, we prove that the complex opposite each chamber in a spherical building Δ \Delta contains an apartment, provided Δ \Delta is thick enough and A u t ( Δ ) Aut(\Delta ) acts chamber transitively on Δ \Delta .

show abstract

On the Finiteness length of some soluble linear groups

Cited by 3 publications

References 58 publications

Between buildings and free factor complexes: A Cohen–Macaulay complex for Out(RAAGs)

Between buildings and free factor complexes: A Cohen–Macaulay complex for Out(RAAGs)

Twisted conjugacy in soluble arithmetic groups

The Σ-invariants of 𝑆-arithmetic subgroups of Borel groups

Contact Info

Product

Resources

About