A finitely generated group that does not satisfy the Burghelea Conjecture

0.07 MB·

by  A. Dranishnikov

A finitely generated group that does not satisfy the Burghelea
Conjecture
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A. Dranishnikov and M. Hull
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Abstract
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We construct a finitely generated group that does not satisfy the Burghelea conjecture.
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T 1 Burghelea Conjecture
K
.
h For an element x in a group G we denote by G the centralizer of x.
x
t
a
Computation of the periodic cyclic homology of a group algebra QG was performed in [2].
m
[
PHC∗(QG) = M H2n+∗(Nx,Q)⊕ M T∗(x,Q)
v1 [x]∈hGifin,n≥0 hGi∞
5
6 where the group T∗(x,Q) fits into the following short exact sequence
1
3 0 → lim1{H (N ,Q)} → T (x,Q) → lim{H (N ,Q)} → 0.
∗−1+2n x ∗ ∗+2n x
0 ← ←
.
1
fin
0 Here Nx = Gx/hxi is the reduced centralizer, hGi denotes the set of conjugacy classes on G, hGi
7 is the set of conjugacy classes of elements of finite order, and hGi∞ the set of conjugacy classes of
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elementsofinfiniteorder. ThebondingmapsintheinversesequencesaretheGysinhomomorphisms
:
v S : H (N ;Q) → H (N ;Q) corresponding to the fibration S1 → BG → BN .
m+2 x m x x x
i
X
Conjecture 1.1 (Generalized Burghelea Conjecture). Let G be a discrete group, then T (x,Q) = 0
r ∗
a for all x ∈hGi∞.
Burghelea stated the above conjecture for geometrically finite groups [2]. In the same paper he
constructed a countable group that does not satisfy the Generalized Burghelea Conjecture.
Theorem 1.2. There is a finitely generated group G that does not satisfy the Generalized Burghelea
Conjecture.
Ourstrategyistoshowthatatorsion-freeversionofthecountablegroupconstructedbyBurghe-
lea can be embedded in a finitely generated group in a way that preserves centralizers. This em-
bedding is based on the theory of small cancellation over relatively hyperbolic groups developed by
Osin [5].
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2 Malnormal Embeddings
ForagroupGhyperbolicrelative toasubgroupH,asubgroupS ofGiscalled suitable ifS contains
two infinite order elements f and g which are not conjugate to any elements of H such that no
non-trivial power of f is conjugate to a non-trivial power of g.
The following is a simplification of [5, Theorem 2.4].
Theorem 2.1. Let G be a torsion-free group hyperbolic relative to a subgroup H, let t ∈ G, and let
S be a suitable subgroup of G. Then there exists a group G and an epimorphism γ: G → G such
that
1. γ| is injective.
H
2. G is hyperbolic relative to H.
3. γ(t) ∈ γ(S).
4. γ(S) is a suitable subgroup of G.
5. G is torsion-free.
We inductively apply the previous theorem to construct the desired embedding. This can be
extracted from the proof of [5, Theorem 2.6], but since it is not explicitly stated there we include
the proof below.
RecallthatasubgroupH ofagroupGiscalledmalnormal ifforallx ∈ G\H,x−1Hx∩H = {1}.
Theorem 2.2. Let C be a torsion-free countable group. Then there exists a finitely generated group
Γ which contains C as a malnormal subgroup.
Proof. Let C = {1 = c ,c ,c ,...}. We inductively define a sequence of quotients as follows: Let
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G = C∗F, where F = F(x,y) is the free group on {x,y}. Then G is hyperbolic relative to C and
0 0
F is a suitable subgroup of G . Suppose now we have constructed a torsion-free group G together
0 i
with an epimorphism α : G ։ G such that
i 0 i
1. α | is injective (so we identify C with its image in G )
i C i
2. G is hyperbolic relative to C.
i
3. α (c ) ∈α (F) for 1 ≤ j ≤i.
i j i
4. α (F) is a suitable subgroup of G .
i i
5. G is torsion-free.
i+1
GivensuchG ,wecanapplyTheorem2.1withS = α (F)andt = c . LetG bethequotient
i i i+1 i+1
provided by Theorem 2.1, that is G = G . Define α = γ ◦α , where γ is the epimorphism
i+1 i i+1 i
given by Theorem 2.1. The conditions of Theorem 2.1 imply that G satisfies conditions (1)-(5).
i+1
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Let Γ be the direct limit of the sequence G ։ G ։ G ..., that is Γ = G / ker(α ). Let
0 1 2 0 S i
β: G → Γ be the natural quotient map. Note that β| is surjective by construction. Indeed, G
0 F 0
is generated by C ∪ {x,y} and for each c ∈ C, α (c ) ∈ α (F), hence β(c) ∈ β(F). Thus Γ is
i i i i
generated by {β(x),β(y)}.
Now β| is injective, so C embeds in Γ; we identify C with its image in Γ. Suppose x ∈ Γ such
C
that x−1Cx∩C 6= {1}. Then there exist g,h ∈ C \{1} such that x−1gxh−1 = 1. Let x˜ ∈ G such
0
thatβ(x˜)= x. Thenforsomei ≥ 1,x˜−1gx˜h−1 ∈ kerα . Thismeansthatα (x˜)−1Cα (x˜)∩C 6= {1}.
i i i
Since G is hyperbolic relative to C, C is malnormal in G by [5, Lemma 8.3b]. Hence α (x˜) ∈ C,
i i i
which means that x = β(x˜) ∈ C. Therefore C is malnormal in Γ.
3 Modifications of Kan-Thurston Theorem
Baumslag, Dyer, and Heller [1] made the following addendum to the Kan-Thurston [3] theorem.
Theorem 3.1. For every connected finite simplicial complex L there is a geometrically finite group
G and a map t :K(G,1) → L that induces an isomorphism of cohomology.
Theproofin[4]oftheKan-Thurson-Baumslag-Dyer-Heller theoremcanbeextendedtoproduce
the following:
Theorem 3.2. For every connected locally finite simplicial complex L presented as a union of
finite subcomplexes L = ∪L there is a locally finite complex K = K(G,1) and a proper map
i
t : K(G,1) → L that induces an isomorphism of cohomology and such that K = t−1(L )= K(G ,1)
i i i
for geometrically finite G and the inclusion K → K induces a monomorphism G → G.
i i i+1 i
Proof of Theorem 1.2. Let L be a locally finite complex representing K(Z,2). Let t :K(G,1) → L
be as in Theorem 3.2. We note that lim {H (L;Q),S} =6 0 where S : H (L;Q) → H (L;Q)
← 2n 2n+2 2n
is the Gysin homomorphism for the canonical S1-bundle S∞ → CP∞. Since G = ∪G is the union
i
of geometrically finite groups G , it is torsion free. Let
i
1 → Z = hxi → C → G → 1
be the central extension extension that corresponds to a generator a ∈ H∗(L) = Z[a], deg(a) = 2.
We apply Theorem 2.2 to obtain a malnormal embedding C → Γ into a finitely generated group.
Then Γ = C and N = Γ /hxi = G. Then T(x;Q)6= 0 in the group Γ.
x x x
References
[1] G. Baumslag, E. Dyer and A. Heller, The topology of discrete groups, J. Pure Appl. Algebra
16 (1980), 1-47.
[2] D. Burghelea, The cyclic homology of the group rings, Comment. Math. Helvetici 60 (1985),
354-365.
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[3] D. M. Kan and W. P. Thurston, Every connected space has the homology of a K(π,1),
Topology 15, (1976), 253-258.
[4] C.R.F. Mauder, A Short Proof of a Theorem of Kan and Thurston, Bull. London Math. Soc.
(1981) 13 (4): 325-327.
[5] D. Osin,Smallcancellations over relatively hyperbolicgroupsandembeddingtheorems, Ann.
of Math. 172 (2010), no. 1, 1-39.
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