Relations between counting functions on free groups and free monoids

Hartnick, Tobias; Talambutsa, Alexey

doi:10.4171/ggd/476

Cited by 6 publications

(21 citation statements)

References 23 publications

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“…What Abért calls a pseudocharacter, we will call a homogeneous quasimorphism. This definition is in line with a lot of the literature, in particular it is in line with the articles [21] and [22], which form the basis of this article. In Section 2 we will explain how we can identify the space of homogeneous quasimorphisms on the free group Q(F n ) up to homomorphisms with the second bounded cohomology H 2 b (F n , R).…”

Section: Introductionsupporting

confidence: 67%

“…Remark 3.5. The space C (F n , S) is defined a bit differently in [22]. But both definitions are equivalent.…”

Section: Counting Functions and Counting Quasimorphisms (Up To Bounde...mentioning

confidence: 99%

“…While this gives us some intuition, at this stage we do not even have a way to decide if two weighted subtrees of T n represent the same equivalence class. Hartnick and Talambutsa solve this problem in [22]:…”

Section: 3mentioning

confidence: 99%

“…) by applying the algorithm to [f −g] ∈ C (F n , S). One major step towards this algorithm in [22] is the introduction of a condition on f that guarantees [f ] = 0. In this subsection we will show that the condition on f in [22] even guarantees that f is reduced.…”

Section: 3mentioning

confidence: 99%

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Dynamics of $\mathrm{Out}(F_n)$ on the second bounded cohomology of $F_n$

Hase

2018

Preprint

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We study the Out(Fn)-action on the second bounded cohomology H 2 b (Fn, R), focusing on the countable-dimensional dense invariant subspace given by Brooks quasimorphisms. We show that this subspace has no finitedimensional invariant subspaces, in particular no fixpoints, partially answering a question of Miklós Abért. To this end we introduce a notion of speed of an element g ∈ Out(Fn), which measures the asymptotic growth rate of bounded cohomology classes under repeated application of g. 1 Definition 2.1. We call a connected topological space BG an Eilenberg-MacLane space of a group G if π 1 (BG) = G and π k (BG) = 0 for all k > 1.Proposition 2.2. For every group G there exists an Eilenberg-MacLane space BG. It is unique up to weak homotopy equivalence.Proof. See [16] and [17] for the original proof. Most textbooks on algebraic topology also give a proof, for example [23] in Section 4.2.Definition 2.3. The group cohomology of G with coefficients in R is defined as the cohomology of the Eilenberg-MacLane space BG with coefficients in R, i.e.a map on the level of cohomology:One of the main results in the study of bounded cohomology is the fact that the comparison map is in general neither injective nor surjective. It is easy to see in an example that c • does not have to be surjective.Example 2.26. The map c 1 :To find an example in which c • is not injective is harder. What helps us is that we can find a nice description of the kernel of c 2 . Here quasimorphisms enter the picture.D(f ) is called the defect of the quasimorphism f . We denote by Q(G) the space of quasimorphisms on a group G. We define an equivalence relation on Q(G) byInstead of looking at equivalence classes of quasimorphisms, one can also pick a suitable representative for every equivalence class and consider the space of representatives. Definition 2.28. A quasimorphism is called homogeneous if for every g ∈ G and n ∈ N we have f (g n ) = nf (g).Proposition 2.29. In every equivalence class [f ] ∈ Q(G) there is exactly one homogeneous quasimorphism f . So we can identify Q(G) with the space of homogeneous quasimorphisms.Proof. We prove the proposition in four steps:Example 3.11. Again let f = 5#aa − 3#ab + #b. We have f S = 2. Both the vertex aa and ab are of distance 2 from the root, the vertex b is of distance 1.3.2. Counting functions up to bounded distance.Definition 3.12. We define an equivalence relation on C (F n , S) byTHE Out(Fn)-ACTION ON H 2 b (Fn, R) 13

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Section: Introductionsupporting

confidence: 67%

“…Remark 3.5. The space C (F n , S) is defined a bit differently in [22]. But both definitions are equivalent.…”

Section: Counting Functions and Counting Quasimorphisms (Up To Bounde...mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

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Dynamics of $\mathrm{Out}(F_n)$ on the second bounded cohomology of $F_n$

Hase

2018

Preprint

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“…In this case we know due to a result of Grigorchuk [Gri95] that the Brooks counting quasimorphisms generate a dense subspace of H 2 b (F ; R) (albeit only in the non-complete topology of pointwise convergence of homogeneous representatives). The linear relations between arbitrary Brooks counting quasimorphisms in H 2 b (F ; R) and various explicit bases for the dense subspace they generate are known [HT15], and there are some very modest beginnings of a study of the structure of H 2 b (F ; R) as an Out(F )-module [HS16,Has16]. Nothing comparable is known for any other groups, not even surface groups.…”

Section: Introductionmentioning

confidence: 99%

Bounded cohomology and virtually free hyperbolically embedded subgroups

Hartnick

Sisto

2019

Groups Geom. Dyn.

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Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group G (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, Out(Fn), . . . ) embeds via the natural restriction maps into the inverse limit of the second bounded cohomologies of its virtually free subgroups, and in fact even into the inverse limit of the second bounded cohomologies of its hyperbolically embedded virtually free subgroups. This result is new and non-trivial even in the case where G is a (non-free) hyperbolic group. The corresponding statement fails in general for the third bounded cohomology, even for surface groups.

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Median quasimorphisms on $${{\,\mathrm{{CAT}}\,}}(0)$$ cube complexes and their cup products

Brück,

Fournier-Facio,

Löh

2023

Geom Dedicata

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Cup products provide a natural approach to access higher bounded cohomology groups. We extend vanishing results on cup products of Brooks quasimorphisms of free groups to cup products of median quasimorphisms, i.e., Brooks-type quasimorphisms of group actions on $${{\,\mathrm{{CAT}}\,}}(0)$$ CAT ( 0 ) cube complexes. In particular, we obtain such vanishing results for groups acting on trees and for right-angled Artin groups. Moreover, we outline potential applications of vanishing results for cup products in bounded cohomology.

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Relations between counting functions on free groups and free monoids

Cited by 6 publications

References 23 publications

Dynamics of $\mathrm{Out}(F_n)$ on the second bounded cohomology of $F_n$

Dynamics of $\mathrm{Out}(F_n)$ on the second bounded cohomology of $F_n$

Bounded cohomology and virtually free hyperbolically embedded subgroups

Median quasimorphisms on $${{\,\mathrm{{CAT}}\,}}(0)$$ cube complexes and their cup products

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