On topological complexity of hyperbolic groups

TL;DR

This paper establishes that the topological complexity of a finitely generated torsion free hyperbolic group is exactly twice its cohomological dimension, providing a precise measure of its topological intricacy.

Contribution

It proves a exact formula linking the topological complexity and cohomological dimension for a class of hyperbolic groups, which was previously unknown.

Findings

Topological complexity equals twice the cohomological dimension for these groups.

Provides a new exact calculation method for topological complexity.

Enhances understanding of the relationship between algebraic and topological properties of hyperbolic groups.

Abstract

We show that the topological complexity of a finitely generated torsion free hyperbolic group $\pi$ with $\cd\pi=n$ equals $2n$.

Equations

$$f:EG\to E_{\mathcal{F}}G$$

$$(\ast )H_{\gamma ,b}=(\gamma ,e)\Delta (Z(b))({\gamma }^{-1},e)$$

$${\mathcal{D}}^{\prime }=\coprod _{a\in A}\{g\Delta (Z(a))g^{-1}\mid g\in G\}$$

$$(x,y)\Delta (Z(a))(x^{-1}{y}^{-1})=(x{y}^{-1},e)\Delta (yZ(a)y^{-1})((x{y}^{-1}{)}^{-1},e),$$

$$(\gamma ,e)(b^k,b^k)({\gamma }^{-1},e)=(\gamma x,x)(a^{\ell },a^{\ell })((\gamma x{)}^{-1},x^{-1})\in (\gamma x,x)\Delta (Z(a))(\gamma x,x{)}^{-1}.$$

$$H_{\gamma ,b}\in \{g\Delta (Z(a))g^{-1}\mid g\in G\}.$$

$$g\Delta (Z(a))g^{-1}=h\Delta (Z(b))h^{-1}.$$

$$g^{-1}h\Delta (Z(b))h^{-1}g\subset \Delta (\pi )$$

$$(x,y)\Delta (Z(b))(x^{-1},y^{-1})=(x,x)(e,b^k)\Delta (Z(b))(e,b^{-k})(x^{-1},x^{-1})=$$

$$\Delta (xZ(b)x^{-1})=\Delta (Z(xb{x}^{-1}))=\Delta (Z(a)).$$

$$Z=X{\cup }_{\bigcup g\varphi }(\coprod _{g\in G}{D}^n)$$

$$X=(E\pi \times E\pi )/\{g\Delta {\}}_{g\in G}$$

$$\{g(a^n,e)(v)\mid n\in \mathbb{Z}\}$$

$$(a^n,e)\Delta (Z(a))(a^{-n},e)=\Delta (Z(a)).$$

$$\varphi :\partial B\to D\cup (a,a)D\cup X^{(2)}.$$

$$X_a=\bigcup _{[g]\in G/\Delta (Z(a))}g{Y}_a$$

$$X_3=\bigcup _{a\in A}{X}_a$$

$$X_3\setminus (\bigcup _{a\in A}G{R}_a).$$

$$B_{\mathcal{D}}G=\bigcup _{a\in A}{X}_a/G=(B\pi \times B\pi )/\Delta (B\pi )\cup \bigcup _{a\in A}(e^1\cup e^2\cup e^3)$$

$$(B\pi \times B\pi )/\Delta (B\pi )\cup \bigcup _{a\in A}{e}^3$$

$$B_{\mathcal{D}}G\sim (B\pi \times B\pi )/\Delta (B\pi )\vee \underset{a\in A}{\bigvee }{S}^3.$$

$$H^{2n}(\pi \times \pi ,\mathbb{Z}(\pi \times \pi ))\ne 0.$$

$$H^k(G,\mathbb{Z}G)=H_c^k(EG;\mathbb{Z})$$

$$Z=\bigcup _{g\in G}g\Delta (E\pi )\cup E^{(2)}.$$

$${\bar{q}}^{\ast }:H^{2n}(\alpha E/\alpha Z;\mathbb{Z})\to H^{2n}(\alpha E;\mathbb{Z}).$$

$$X_3\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\xi \circ \psi \circ \Phi :E\to \alpha E/\alpha Z$$

$$\xi \psi \Phi (x_n\times I)\cap (\alpha E/\alpha Z)\setminus U\ne \varnothing$$

$$C={\xi }^{-1}((\alpha E/\alpha Z)\setminus U)$$

$$\bar{p}:E/Z\to B/p(Z)$$

Full text

1. Introduction

The topological complexity of a space is a numerical invariant defined by M. Farber [F] in his study of motion planning in robotics. The topological complexity ${\mathop{\mathrm{missing}}{TC}\,}(X)$ of a space $X$ is the minimum of $k$ such that $X\times X$ admits an open cover $U_{0},\dots,U_{k}$ with the property that over each $U_{i}$ there is a continuous motion planning algorithm $s_{i}$, i.e. a continuous map $s_{i}:U_{i}\to PX$ to the path space $PX=X^{[0,1]}$ such that $s_{i}(x,y)(0)=x$ and $s_{i}(x,y)(1)=y$ for all $(x,y)\in U_{i}$.

The topological complexity is a homotopy invariant in spirit of the Lusternik-Schnirelmann category $\operatorname{cat}(X)$. Like in the case of the LS-category, generally this invariant is hard to compute. Since ${\mathop{\mathrm{missing}}{TC}\,}(X)$ is homotopy invariant, one can define ${\mathop{\mathrm{missing}}{TC}\,}(\pi)$, the topological complexity of a discrete group $\pi$ as ${\mathop{\mathrm{missing}}{TC}\,}(\pi)={\mathop{\mathrm{missing}}{TC}\,}(B\pi)$. This definition works for the LS-category as well. Both group invariants turns to be $\infty$ in the presence of torsions. So we consider torsion free groups only. Eilenberg and Ganea [EG] proved that the LS-category of a discrete group equals its cohomological dimension, $\operatorname{cat}(\pi)=\operatorname{cd}(\pi)$. It turns out that in the case of topological complexity the range of ${\mathop{\mathrm{missing}}{TC}\,}(\pi)$ is between $\operatorname{cd}(\pi)$ and $2\operatorname{cd}(\pi)$ (see [CP],[Ru],[DS]).

The minimal value of ${\mathop{\mathrm{missing}}{TC}\,}$ is taken on abelian groups, ${\mathop{\mathrm{missing}}{TC}\,}(\pi)=\operatorname{cd}(\pi)$. Combining results of  [DS] and [Dr1] present the cases of the maximal value ${\mathop{\mathrm{missing}}{TC}\,}(\pi)=2\operatorname{cd}(\pi)$ for groups which are free ”square”, $\pi=H\ast H$, of a geometrically finite group. In this paper we investigate further the question when the topological complexity of a group is maximal. We consider groups $\pi$ with finite classifying complex $B\pi$. In particular, in this paper we prove the equality ${\mathop{\mathrm{missing}}{TC}\,}(\pi)=2\operatorname{cd}(\pi)$ for hyperbolic groups. A giant step in computation of ${\mathop{\mathrm{missing}}{TC}\,}$ for hyperbolic groups was made by Farber and Mescher [FM]. In the light of the formula $\operatorname{cd}(\pi\times\pi)=2\operatorname{cd}(\pi)$ [Dr1] their result states that for hyperbolic groups with $\operatorname{cd}(\pi)=n$ either ${\mathop{\mathrm{missing}}{TC}\,}(\pi)=2n$ or ${\mathop{\mathrm{missing}}{TC}\,}(\pi)=2n-1$.

In the paper we use the following properties of torsion free hyperbolic groups: The centralizer of every nontrivial element is cyclic and a finitely generated torsion free hyperbolic group $\pi$ has finite classifying complex $B\pi$. Our main tool is the characterization of ${\mathop{\mathrm{missing}}{TC}\,}(\pi)$ given by Farber-Grant-Lupton-Oprea [FGLO]:

1.1 Theorem**.**

The topological complexity ${\mathop{\mathrm{missing}}{TC}\,}(\pi)$ of a group $\pi$ with finite $B\pi$ equals the minimal number $k$ such that the canonical map $f:E(\pi\times\pi)\to E_{\mathcal{D}}(\pi\times\pi)$ admits a $(\pi\times\pi)$-equivariant deformation to the $k$-skeleton $E_{\mathcal{D}}(\pi\times\pi)^{(k)}$.

Here $E_{\mathcal{F}}G$ is the notations for a classifying space for $G$-action with isotropy groups in the class of subgroups $\mathcal{F}$ [Lu]. The class $\mathcal{D}$ is defined by conjugations of the diagonal subgroup $\Delta(\pi)\subset\pi\times\pi$ and their intersections.

It turns out to be that the essentiality of $f$ in the top dimension can be detected cohomologically. In the case when $B\pi$ is an oriented manifold this can be done using the induced map of the orbit spaces $\bar{f}:B(\pi\times\pi)\to B_{\mathcal{D}}(\pi\times\pi)$. Generally, our argument uses compactifications of the classifying spaces and the main result of [Dr1].

1.1. Notations and conventions

In this paper a group $\pi$ that admits a finite classifying space $B\pi$ is called geometrically finite. Hyperbolic groups in the paper a Gromov’s hyperbolic [Gr].

The notations for cohomology of a group is $H^{*}(\pi,M)$ and for cohomology of a space $H^{*}(X;M)$.

We consider an actions of a discrete group $G$ on CW-complex $X$ that preserve the CW-complex structure in such a way that if an open cell $e$ is taken by an element $g\in G$ to itself then $g$ restricted to $e$ is the identity map. Such a complex is called a $G$-CW-complex.

2. Classifying space $E_{\mathcal{D}}G$

For a discrete group $G$ and a family of its subgroups $\mathcal{F}$ which is closed under conjugation and finite intersection a classifying complex $E_{\mathcal{F}}G$ is a $G$-CW-complex whose isotropy subgroups are from $\mathcal{F}$ and for each $G$-CW complex with isotropy groups from $\mathcal{F}$ there is a uniqe up to a $G$-homotopy a $G$-equivariant map $f:X\to E_{\mathcal{F}}G$. This implies in particular,that any two such complexes a $G$-homotopy equivalent. Note that the universal cover $EG$ of the classifying space $BG$ is a $G$-CW-complex. The corresponding map

[TABLE]

is called a canonical map.

It was proven in [Lu] that a $G$-CW-complex $X$ is $E_{\mathcal{F}}G$ if and only if for each group $F\in\mathcal{F}$ the fixed point set $X^{F}$ is contractible.

2.1. The family $\mathcal{D}$ for groups with cyclic centralizers

Let $\pi$ be a torsion free discrete group, we set $G=\pi\times\pi$. We denote by $\mathcal{D}$ the smallest family of subgroups $H\subset G$ which contains the diagonal $\Delta(\pi)\subset\pi\times\pi$, the trivial subgroup and which is closed under conjugations and finite intersections. A complete description of the family $\mathcal{D}$ is given in 4.1 of [FGLO]. Let $\mathcal{D}^{\prime}$ denote the set of all groups in $\mathcal{D}$ except the trivial group.

Also it was proven in [FGLO] that for a torsion free group $\pi$ with cyclic centralizers for any two centralizers $Z(a)$, $Z(b)$ of elements $a,b\in\pi$ either $Z(a)=Z(b)$ or $Z(a)\cap Z(b)=\{e\}$. Their description of $\mathcal{D}$ in the case of cyclic centralizers can be stated as follows:

2.1 Proposition**.**

For a torsion free group $\pi$ with cyclic centralizers a subgroup $H\subset\pi\times\pi$ belongs to $\mathcal{D}^{\prime}$ if and only if it is of the form

[TABLE]

where $\gamma,b\in\pi$ and $\Delta:\pi\to\pi\times\pi$ is the diagonal map.

Thus, the family $\mathcal{D}^{\prime}$ for a torsion free group with cyclic centralizers consists of the conjugates of the diagonal subgroup $\Delta(\pi)\subset\pi\times\pi$ and the conjugates of the centralizers of all nontrivial elements in $\Delta(\pi)$.

2.2 Proposition**.**

Given $\gamma,b,\alpha,c\in\pi$, either $H_{\gamma,b}=H_{\alpha,c}$ or $H_{\gamma,b}\cap H_{\alpha,c}=(e,e)$.

Proof.

If $(e,e)\neq(x,y)\in H_{\gamma,b}\cap H_{\alpha,c}$, then $c\in Z(b)$ and hence $H_{\alpha,c}=H_{\alpha,b}$. Also, $\gamma b^{k}\gamma^{-1}=\alpha b^{k}\alpha^{-1}$ for some $k$. Hence $\alpha^{-1}\gamma\in Z(b)$ and hence, $\gamma=\alpha b^{n}$ for some $n$. Then $H_{\gamma,b}=H_{\alpha,b}$. ∎

We call two elements $a,b\in\pi$ weakly conjugate if there are integers $k$ and $\ell$ such that $a^{k}$ is conjugate to $b^{\ell}$. Clearly this is an equivalence relation. Note that for a torsion free group the unit form a separate equivalence class. Let $C$ be the set of the weak conjugacy classes of nontrivial elements in $\pi$. Let $A$ be a section of $C$, i.e. a collection of elements $a_{F}\in F\in C$ indexed by $C$. We may assume that $a$ is a generator of $Z(a)\cong{\mathbb{Z}}$. If not, then $Z(a)$ is generated by some $t$ and $a=t^{n}$ lies in the same weak conjugacy class. So we can choose $t$ instead of $a$.

2.3 Proposition**.**

The set of groups $\mathcal{D}^{\prime}$ can be presented as the disjoint union

[TABLE]

of families of groups indexed by $a\in A$ where each family consists of all distinct conjugates of $\Delta(Z(a))$.

Proof.

Note that every conjugate

[TABLE]

$(x,y)\in\pi\times\pi=G$, is one of the groups $H_{\gamma,b}$ from Proposition 2.1. Suppose that $b$ belongs to the weak conjugacy class of $a$, $b^{k}=xa^{\ell}x^{-1}$. Then

[TABLE]

By Proposition 2.2 $H_{\gamma,b^{k}}$ coincides with $(\gamma x,x)\Delta(Z(a))(\gamma x,x)^{-1}$. Since $Z(b)=Z(b^{k})$, we obtain $H_{\gamma,b}=H_{\gamma,b^{k}}$ and hence,

[TABLE]

Now we show that the families in the union are disjoint. Suppose that

[TABLE]

Then $\Delta(Z(a))=g^{-1}h\Delta(Z(b))h^{-1}g$. Let $g^{-1}h=(x,y)$. Then the condition

[TABLE]

implies $xbx^{-1}=yby^{-1}$. Therefore, $x^{-1}y\in Z(b)$ and, hence, $y=xb^{k}$ for some $k$. Then

[TABLE]

[TABLE]

The last equality implies that $xbx^{-1}=a^{n}$ for some $n$. Thus, $b$ is weakly equivalent to $a$. ∎

We note that the unit $e\in\pi$ form a separate conjugacy class with $Z(e)=\pi$. Thus, the summand corresponding to $e\in A$ consists of conjugates of the diagonal subgroup.

2.2. Construction of the classifying space $E_{\mathcal{D}}G$

Assume that a group $G$ acts on a space $X$ and $A$ is a $G$-invariant closed subset. Let $\phi:A\to Y$ be a $G$-equivariant map. Then, clearly, the space $Y\cup_{\phi}X$ obtained from attaching $X$ to $Y$ along $A$ admits a natural $G$-action.

2.4 Definition**.**

Let $X$ be a CW-complex with a $G$-action preserving the CW-complex structure. Let $\phi:\partial D^{n}\to X^{(n-1)}$ be an attaching map. We say that the complex

[TABLE]

is obtained from $X\cup_{\phi}D^{n}$ by the $G$-translation. Thus, the action of $G$ on $X$ extends to $Z$.

Let $G=\pi\times\pi$ and let $E\pi$ be the universal covering of a classifying complex $B\pi$ of the group $\pi$. As the above we assume that $\pi$ is torsion free and it has centralizers of all nontrivial elements cyclic. Note that the translates of the diagonal $\Delta=\Delta(E\pi)\subset E\pi\times E\pi$ by elements of $g\in G$ are either disjoint or coincide. We consider the quotient space

[TABLE]

obtained from the product $E\pi\times E\pi$ by collapsing to points all translates of the diagonal. We may assume that $EG$ has a $G$-equivariant CW-complex structure such that $\Delta$ is a subcomplex. Then $X$ has the quotient CW-complex structure. Note that the distinct translates $g\Delta$ of $\Delta$ can be indexed by elements of $\pi$ in the following way $\{(\gamma,e)\Delta\}_{\gamma\in\pi}$. We denote the image of of $(\gamma,e)\Delta$ in $X$ by $v_{\gamma}$ and the image of $\Delta$ by $v=v_{e}$. Thus, $v_{\gamma}=(\gamma,e)v$ are vertices in $X$. Note that $X$ is contractible and the action of $G$ on $E\pi\times E\pi$ induces a $G$-action on $X$.

Let $a$ be a generator of the centralizer $Z(a)\cong{\mathbb{Z}}$, $a\in\pi$ Then the set $\{(a^{n},e)v\mid n\in{\mathbb{Z}}\}$ is the fixed point set for the group $\Delta(Z(a))$ and the set

[TABLE]

is the fixed point set of the group $g\Delta(Z(a))g^{-1}$ for $g\in G$.

We construct $E_{\mathcal{D}}G$ by attaching to $X$ cells of dimensions 1, 2, and 3.

We consider the natural action of $G$ on the set of unordered pairs $\{\{u,v\}\mid u,v\in Gv\}$.

2.5 Proposition**.**

For any $a\in A$ the stabilizer of the pair $\sigma=\{v,v_{a}\}$ is the centralizer $\Delta(Z(a))$.

Proof.

Suppose that $(x,y)\{v,v_{a}\}=\{v,v_{a}\}$ $x,y\in\pi$. There are two cases: $(x,y)v=v$ and $(x,y)v_{a}=v_{a}$ or $(x,y)v=v_{a}$ and $(x,y)v_{a}=v$.

In the first case, from the equality $(x,y)v=(xy^{-1},v)v=v$ we obtain that $xy^{-1}=e$ and, hence, $x=y$. From the condition $(x,y)v_{a}=v_{a}$ we obtain $(x,y)v_{a}=(x,x)(a,e)v=(xa,x)v=(xax^{-1},e)v=(a,e)v$ and, hence, $xa^{-1}x=a$. Thus $x\in Z(a)$ and $(x,y)\in\Delta(Z(a))$.

In the second case we obtain the pair $(x^{2},y^{2})$ satisfies the conditions of the first case. Therefore, $x^{2}=y^{2}$ and $x^{2},y^{2}\in Z(a)$. The latter implies that $x\in Z(a)$ and $y\in Z(a)$ (see [FGLO]). ∎

2.6 Corollary**.**

For every $n\in{\mathbb{Z}}$ the stabilizer of the pair $\sigma_{n}=\{v_{a^{n}},v_{a^{n+1}}\}$ is $\Delta(Z(a))$.

Proof.

We note that $\sigma_{n}=(a^{n},e)\sigma$. Hence the stabilizer of $\sigma_{n}$ is the conjugate of the stabilizer of $\sigma$,

[TABLE]

∎

For a fixed $a\in A$ we construct a $G$-equivariant CW complex $X_{a}$ containing $X$ as an invariant subset. Let the pair $\sigma=\{v,v_{a}\}$ also denote the interval spanned by $\{v,v_{a}\}$. We consider such an interval for each pair of points in the orbit $G\{v,v_{a}\}$. Thus, we obtain a $G$-equivariant family of intervals attached to $X$ such that no two interval have the same set of vertices. We denote a new complex by $X_{a}^{1}$. The action of $G$ on $X$ extends to $X_{a}^{1}$. Note that in $X^{1}_{a}$ the intervals $(a^{n},e)\sigma$, $n\in{\mathbb{Z}}$, form a real line $R_{a}\cong{\mathbb{R}}$ attached to $X$ along the integers ${\mathbb{Z}}\subset{\mathbb{R}}$ to the vertices $v_{a^{n}}$. Then $R_{a}$ is the fixed point set for $\Delta(Z(a))$ and it is contractible. Moreover, each group $g\Delta(Z(a))g^{-1}$, $g\in G$ has the contractible fixed point set $gR_{a}$.

Next we will make $X_{a}^{1}$ simply connected. For that we choose a subcomplex $I\subset X$ homeomorphic to the interval $[0,1]$ connecting the vertices $v$ and $v_{a}$ and attach a 2-disk $D$ to the circle $I\cup\sigma$. Then we define $X_{a}^{2}$ to be a $G$-complex obtained from $X_{a}^{1}\cup D$ by the $G$-translation.

The disks $D$ and $(a,a)D$ together with a contractible space $X$ form in $X_{a}^{2}$ a unique up to homotopy 2-spheroid $S_{n}$. We fill it with a 3-ball $B$, using an attaching map for

[TABLE]

We define a $G$-complex $X_{a}$ as the complex obtained from $X^{2}_{a}\cup_{\phi}B$ by the $G$-translation.

2.7 Proposition**.**

The CW complex $X_{a}$ has the following properties:

$X_{a}$* is $G$-equivariant;*

• 2.

$X_{a}$* is contractible;*

• 3.

The groups $g\Delta(Z(a))g^{-1}$ have contractible fixed point sets;

• 4.

$\dim(X_{a}\setminus X)=3$.

Proof.

The conditions 1, 3, and 4 are obvious. We prove 2.

Let $Y_{a}$ be a CW-complex obtained by the above procedure from $X$ by attaching the cells $\sigma$, $D$, and $B$ with the translations by the group $\Delta(Z(a))\cong{\mathbb{Z}}$. Since $\sigma$ is fixed by $\Delta(Z(a))$, the complex $Y_{a}$ has one new 1-cell $\sigma$. In dimensions 2 and 3, it has cells indexed by ${\mathbb{Z}}$ such that the quotient CW complex $Y_{a}/(X\cup D)$ isomorphic to the reduced double suspension over the reals ${\mathbb{R}}$ with ${\mathbb{Z}}$ as the 0-skeleton. Therefore, $Y_{a}$ is contractible. Note that

[TABLE]

with $X$ being the common part of all summands $gY_{a}$. Hence $X_{a}$ is contractible. ∎

We perform this construction for all $a\in A$ and define

[TABLE]

which is obtained from the disjoint union $X_{3}=\coprod_{a\in A}X_{a}$ by the identification of all copies of $X\subset X_{a}$.

2.8 Proposition**.**

For $a,b\in A$, $a\neq b$, The orbits of the pairs $G\{v,v_{a}\}$ never lands in $R_{b}$.

Proof.

Suppose $g(v)=v_{b^{k}}$ and $g(v_{a})=v_{b^{m}}$. Let $g=(x,y)\in\pi\times\pi$. The first equality implies that $(x,y)v=(xy^{-1},e)v=(b^{k},e)v$ and, hence, $x=b^{k}y$. The second equality implies $(x,y)(a,e)v=(xa,y)v=(xay^{-1},e)v=(b^{m},e)v$ and, therefore, $xay^{-1}=b^{m}$. Thus, $b^{k}yay^{-1}=b^{m}$ which implies that $a$ and $b$ are weakly conjugate. This contradicts to the choice of $a$ and $b$. ∎

2.9 Proposition**.**

The complex $X_{3}$ satisfies all the conditions of $E_{\mathcal{D}}G$.

Proof.

The CW complex $X_{3}$ is a $G$-complex as the union of $G$-complexes with a $G$-invariant intersection. Note that $G$ acts freely on

[TABLE]

Already we know that the vertices of the fixed point complex of $\Delta(Z(b))$ are those of $R_{b}$. In view of Proposition 2.8, no new edges are spanned by vertices in $R_{b}$ and hence the fixed point set of $\Delta(Z(b))$ is just $R_{b}$ which is contractible. Therefore, the fixed point sets are contractible for all groups $H\in\mathcal{D}$.

The complex $X_{3}$ is contractible as a union of contractible complexes $X_{a}$ with a contractible intersection. ∎

2.3. CW structure of $B_{\mathcal{D}}G$

Note that $B_{\mathcal{D}}G=E_{\mathcal{D}}/G$ contains $(B\pi\times B\pi)/\Delta(B\pi)$ as a subcomplex with a 3-dimensional complement. Moreover, as a CW complex

[TABLE]

where $e^{1}$ is the image of $Int(\sigma)$, $e^{2}$ is the image of $Int(D)$, and $e^{3}$ is the image of the interior of the 3-ball $B$ from $X_{a}$. Since $D$ can be deformed to $I$, the complex $B_{\mathcal{D}}G$ is homotopy equivalent to

[TABLE]

where 3-cells are attached to the image of $I$. Hence the attaching maps are null-homotopic. Thus, the space $B_{\mathcal{D}}G$ is homotopy equivalent to the bouquet of $(B\pi\times B\pi)/\Delta(B\pi)$ with a wedge of 3-spheres,

[TABLE]

3. Conditions when ${\mathop{\mathrm{missing}}{TC}\,}(\pi)$ is maximal

We use the following

3.1 Theorem** ([Dr1]).**

For a geometrically finite group $\pi$ with $\operatorname{cd}(\pi)=n$,

[TABLE]

We note that for a group $G$ with finite $BG$ there is the equality [Bro]

[TABLE]

where $EG$ is the universal cover of $BG$ and $H_{c}^{*}$ stands for cohomology with compact supports.

Here is our main result.

3.2 Theorem**.**

Suppose that a geometrically finite group $\pi$ has cyclic centralizers and $\operatorname{cd}\pi=n\geq 2$. Then ${\mathop{\mathrm{missing}}{TC}\,}(\pi)=2n$.

Proof.

Let $G=\pi\times\pi$ and let $E=E\pi\times E\pi$ where $E\pi$ is the universal cover of a finite simplicial complex $B\pi=E\pi/\pi$. Let $B=BG=E/G$. Suppose that the canonical map $f:E\to X_{3}$ can be deformed by a $G$-homotopy $\Phi:E\times I\to X_{3}$ to a map with the image in $X_{3}^{(2n-1)}$. Let

[TABLE]

We use the notation $\alpha Y$ for the one-point compactification of a locally compact space $Y$. Since $\dim\alpha Z=\max\{n,2\}<2n-1$, the exact sequence of pair implies that $H^{2n}(\alpha E)=H^{2n}(\alpha E,\alpha Z)$. Therefore, the collapsing map $\bar{q}:\alpha E\to\alpha E/\alpha Z$ induces an isomorphism of cohomology groups

[TABLE]

Consider the commutative diagram

[TABLE]

where $\psi$ is defined by sending $X_{3}\setminus X$ to the point in $E/Z$ generated by $Z$. The map $\xi$ is a continuous bijection which is a homeomorphism in the complement of the singular one-point sets defined by $Z$ and $\alpha Z$.

Let $\alpha=\alpha E\setminus E$ denote the point at infinity. We claim that sending $\alpha\times I$ to the singular point $z_{0}\in\alpha E/\alpha Z$ continuously extends the homotopy

[TABLE]

to a continuous homotopy $\hat{\Phi}:\alpha E\times I\to\alpha E/\alpha Z$.

Assume the contrary: There is an open neighborhood $U$ of $z_{0}$ and a sequence $x_{n}\in E$ converging to $\alpha$ such that

[TABLE]

for all $n$. Then $\psi\Phi(x_{n}\times I)\cap C\neq\emptyset$ for all $n$ where

[TABLE]

is a compact subset of $E\setminus Z\subset E/Z$. Fix $t_{n}\in I$ such that $\psi\Phi(x_{n},t_{n})\in C$. Let $p:E\to B$ be the universal covering map and let

[TABLE]

be the induced map. We consider a metric $d_{E}$ on $E$ lifted from a metric on $B$. Let $V$ be an open neighborhood of the singular point $\bar{b}_{0}\in B/p(Z)$ such that

[TABLE]

where $\bar{V}$ is the closure of $V$. We may assume that the quotient map $\nu:B\to B/p(Z)$ is 1-Lipschitz. Consider the metric $d^{\prime}$ on $W=E/Z\setminus\bar{p}^{-1}(V)$ lifted with respect to $\bar{p}$ from the matric on $B/p(Z)\setminus V$. Note that the action of $G$ on $W$ is proper discontinuous by isometries. Let $\epsilon=dist_{d^{\prime}}(C,p^{-1}\partial V)>0$. The homotopy $\psi\Phi$ defines a homotopy $\hat{\Phi}:B\times I\to B/p(Z)$ such that the diagram

[TABLE]

commutes. Then the restriction

[TABLE]

is Lipschitz.

Compactness of $B$ and $I$ implies that there is a subsequence $(x_{n_{k}},t_{n_{k}})$ such that $\{p(x_{n_{k}})\}$ converges to some $\bar{x}\in B$ and $\{t_{n_{k}}\}$ converges to some $t^{*}\in I$. Note that $\bar{\Phi}(\bar{x},t^{*})\in\bar{p}(C)$. Let $p(x)=\bar{x}$. Then $G(x)\times t^{*}\subset W^{\prime}$. By the choice of the metric on $E$ there is a sequence of elements $g_{k}\in G$, $\|g_{k}\|\to\infty$, such that

[TABLE]

Since $\bar{p}(C)\cap\bar{V}=\emptyset$, for sufficiently large $k$ we have $Gx\times t_{n_{k}}\subset W^{\prime}$. Then the Lipschitz condition of the restriction of $\psi\Phi$ to $W^{\prime}$ implies

[TABLE]

for sufficiently large $k$ .

Then

[TABLE]

For sufficiently large $k$ each of the above summands does not exceed $\epsilon/2$. Thus, the sequence

[TABLE]

stays in a bounded distance from a compact set. This contradicts with the properness of the action of $G$ on $W$.

Thus, there is a homotopy

[TABLE]

of $\bar{q}$ to a map with the image in the $(2n-1)$-dimensional compact space $\alpha E^{(2n-1)}/\alpha Z$. Therefore, $\bar{q}$ induces zero homomorphism of $2n$-cohomology. Therefore,

[TABLE]

This contradicts to Theorem 3.1. ∎

4. Maximal value of TC via cohomology of the orbit spaces

In this section we present a shorter proof of the main theorem for Poincare Duality groups $\pi$.

4.1 Proposition**.**

Suppose that for a geometrically finite group $\pi$ with $\operatorname{cd}\pi=n\geq 2$ and cyclic centralizers, $H^{2n}(B\pi\times B\pi;q^{*}\mathcal{A})\neq 0$ for some local system of coefficients $\mathcal{A}$ on the quotient space $(B\pi\times B\pi)/\Delta(B\pi)$ where $q:B\pi\times B\pi\to(B\pi\times B\pi)/\Delta(B\pi)$ is the quotient map. Then ${\mathop{\mathrm{missing}}{TC}\,}(\pi)=2n$.

Proof.

The exact sequence of pairs form a commutative diagram

[TABLE]

where the right vertical homomorphism is an isomorphism in view of excision [Bre]. Thus, $q^{*}$ is an isomorphism. Therefore $q$ cannot be deformed to the $2n-1$-skeleton in $(B\pi\times B\pi)/\Delta(B\pi)$ as well as in $B_{\mathcal{D}}G$. Hence $\tilde{q}:E\pi\times E\pi\to E_{\mathcal{D}}G$ cannot be $(\pi\times\pi)$-equivariantly deformed to the $(2n-1)$-skeleton. By Theorem 1.1, ${\mathop{\mathrm{missing}}{TC}\,}(\pi)\geq 2n$. ∎

4.2 Corollary**.**

Let a group $\pi$ be with cyclic centralizers and with the classifying space $B\pi$ be a manifold. Then ${\mathop{\mathrm{missing}}{TC}\,}(\pi)=2\operatorname{cd}(\pi)$.

Proof.

In the case of orientable manifold $B\pi$ we apply Proposition 4.1 by taking the integers as coefficients.

If $B\pi$ is non-orientable, we consider the tensor product $\mathcal{O}\hat{\otimes}\mathcal{O}$ of the orientation sheaves coming from different factors. Since the action of the diagonal subgroup is trivial on the stalk ${\mathbb{Z}}\otimes{\mathbb{Z}}$, it follows that $\mathcal{O}\hat{\otimes}\mathcal{O}=q^{*}\mathcal{A}$ for some local system $\mathcal{a}$ on $(B\pi\times B\pi)/\Delta(B\pi)$ (see [Dr2]). ∎