The homology of the Higman–Thompson groups

Szymik, Markus; Wahl, Nathalie

doi:10.1007/s00222-018-00848-z

Cited by 15 publications

(35 citation statements)

References 14 publications

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“…The integral cohomology of these groups has been computed [11,26,53], but little is known about their real bounded cohomology. We formulate one question for each group: The bounded cohomology of the Thompson group T is given by…”

Section: Thompson Groups and Their Siblingsmentioning

confidence: 99%

Bounded cohomology and binate groups

Fournier-Facio,

Loeh,

Moraschini

2021

Preprint

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A group is boundedly acyclic if its bounded cohomology with trivial real coefficients vanishes in all positive degrees. Amenable groups are boundedly acyclic, while the first non-amenable examples were the group of compactly supported homeomorphisms of R n (Matsumoto-Morita) and mitotic groups (Löh). We prove that binate (alias pseudo-mitotic) groups are boundedly acyclic, which provides a unifying approach to the aforementioned results. Moreover, we show that binate groups are universally boundedly acyclic.We obtain several new examples of boundedly acyclic groups as well as computations of the bounded cohomology of certain groups acting on the circle. In particular, we discuss how these results suggest that the bounded cohomology of the Thompson groups F , T , and V is as simple as possible.

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Section: Thompson Groups and Their Siblingsmentioning

confidence: 99%

Bounded cohomology and binate groups

Fournier-Facio,

Loeh,

Moraschini

2021

Preprint

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“…A Cantor algebra of arity a is a set X together with a bijection X a → X. The Cantor algebras of arity a are the models for a Lawvere theory Cantor a , and its algebraic K-theory has been computed in [SW19]: K(Cantor a ) S/(a − 1), (5.2)…”

Section: Cantor Algebrasmentioning

confidence: 99%

“…Proof. We start from [SW19], where the algebraic K-theory of Cantor a is identified with the Moore spectrum S/(a − 1). That Moore spectrum is the the cofiber of multiplication with a − 1 on the sphere spectrum, so that K(Z) ∧ K(Cantor a ) is the cofiber of multiplication with a − 1 on K(Z).…”

Section: Cantor Algebrasmentioning

confidence: 99%

“…from the smash product of the algebraic K-theory spectra into it. The homotopy type of this smash product can be worked out, because the algebraic K-theory spectra are Moore spectra by [SW19], but it is not known whether the map (5.3) is an equivalence.…”

Section: Cantor Algebrasmentioning

confidence: 99%

“…Here the theory of abelian groups is given by the modules over the ring Z of integers and Z ⊗ T is the theory of abelian group objects in T , which is always given by modules over a ring. These assembly maps can be equivalences, as happens for Cantor algebras, where the rings in question are the Leavitt algebras (see our Theorem 5.4, based on [SW19]) or they can fail to be even rationally injective, as happens for Boolean algebras (see our Theorem 5.7, based on [BS]), where they are null-homotopic. Specializing further to the theory T of actions by a fixed discrete group G, we recover Loday's assembly map (1.1) for the group G.…”

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confidence: 97%

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Generalizations of Loday's assembly maps for Lawvere's algebraic theories

Bohmann¹,

Szymik²

2021

Preprint

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Loday's assembly maps approximate the K-theory of group rings by the K-theory of the coefficient ring and the corresponding homology of the group. We present a generalization that places both ingredients on the same footing. Building on Elmendorf-Mandell's multiplicativity results and our earlier work, we show that the K-theory of Lawvere theories is lax monoidal. This result makes it possible to present our theory in a user-friendly way without using higher categorical language. It also allows us to extend the idea to new contexts and set up a non-abelian interpolation scheme, raising novel questions. Numerous examples illustrate the scope of our extension.

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Boolean algebras, Morita invariance and the algebraic K-theory of Lawvere theories

Bohmann

Szymik²

2023

Math. Proc. Camb. Phil. Soc.

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The algebraic K-theory of Lawvere theories is a conceptual device to elucidate the stable homology of the symmetry groups of algebraic structures such as the permutation groups and the automorphism groups of free groups. In this paper, we fully address the question of how Morita equivalence classes of Lawvere theories interact with algebraic K-theory. On the one hand, we show that the higher algebraic K-theory is invariant under passage to matrix theories. On the other hand, we show that the higher algebraic K-theory is not fully Morita invariant because of the behavior of idempotents in non-additive contexts: We compute the K-theory of all Lawvere theories Morita equivalent to the theory of Boolean algebras.

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The homology of the Higman–Thompson groups

Cited by 15 publications

References 14 publications

Bounded cohomology and binate groups

Bounded cohomology and binate groups

Generalizations of Loday's assembly maps for Lawvere's algebraic theories

Boolean algebras, Morita invariance and the algebraic K-theory of Lawvere theories

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