\documentstyle[epsf,12pt]{article} \title{The second bounded cohomology \\ of a group with infinitely many ends \footnote{1991 {\it Mathematics Subject Classification}, Primary: 20F32, Secondary: 55N20.}} \author{Koji Fujiwara \thanks{This work was done when the author visited MSRI supported in part by NSF grant DMS-9022140 and a JSPS grant.} \\ Mathematics Department, Keio University \\ Yokohama, 223, Japan \\ {\tt fujiwara@math.keio.ac.jp}} \date{} % \newtheorem{Th}{Theorem} \newtheorem{Prop}{Proposition} \newtheorem{Lemma}{Lemma}[section] \newtheorem{Cor}{Corollary} \newtheorem{Conj}{Conjecture} % \newcommand{\mathbf}{\bf } \newcommand{\qed}{\hspace *{\fill} q.e.d.} \begin{document} \maketitle \begin{abstract} We study the second bounded cohomology of an amalgamated free product of groups, and an HNN extension of a group. As an application, we have a group with infinitely many ends has infinite dimensional second bounded cohomology. \end{abstract} \noindent {\bf Contents} \noindent \S 1 Introduction \noindent \S 2 Quasi homomorphism and the counting function \noindent \S 3 Quasi homomorphisms of $A*_C B$ \noindent \S 4 Choice of words for $A*_C B$ \noindent \S 5 Proofs of Th 1 and Cor 1,2,3 \noindent \S 6 Quasi homomorphisms of $A*_C \varphi $ \noindent \S 7 Choice of words for $A*_C \varphi $ \noindent \S 8 Proofs of Th 2 and Th 3 \section{Introduction} Bounded cohomology was defined by F. Trauber for groups and by M. Gromov for spaces. We review the definition of bounded cohomology of a group $G$. Let $$C^k_b(G;A) = \{ f: G^k \rightarrow A \mid f \mbox{ has bounded range} \}, $$ where $A={\mathbf Z}$ or ${\mathbf R}$. The boundary $\delta:C^k_b(G;A) \rightarrow C^{k+1}_b(G;A)$ is given by \begin{eqnarray*} \delta f(g_0,\ldots,g_k) &=& f(g_1,\ldots,g_k) + \sum_{i=1}^k (-1)^i f(g_0,\ldots,g_{i-1}g_i,\ldots g_k)\\ &&\quad + (-1)^{k+1} f(g_0,\ldots,g_{k-1}). \end{eqnarray*} The cohomology of the complex $\{C^k_b(G;A), \delta \}$ is the {\it bounded cohomology} of $G$, denoted by $H^*_b(G;A)$. See \cite{G}, \cite{I} as general references for the theory of bounded cohomology. For any group $G$, the first bounded cohomology, $H^1_b(G;A)$ is trivial. For an amenable group $G$, $H^n_b(G;{\mathbf R})$ is trivial for any $n$. The first example of a group with non-trivial second bounded cohomology was given by R. Brooks, \cite{B}. He showed a free group of rank greater than 1 has infinite dimensional second bounded cohomology. R.I. Grigorchuk investigated the structure of the second bounded cohomology of free groups, torus knot groups and surface groups, \cite{Gr}. T. Yoshida \cite{Y} and T. Soma \cite{So1}, \cite{So2} studied the third bounded cohomology of surfaces and hyperbolic three manifolds. D.B.A. Epstein and the author showed a non-trivial word-hyperbolic group has infinite dimensional second bounded cohomology, \cite{EF}. We state results on the second bounded cohomology of an amalgamated free product of groups: \begin{Th} Let $G=A*_C B$. If $|C \backslash A/C| \ge 3$ and $ |B/C| \ge 2$, then the cardinality of the dimension of $H^2_b(G;{\mathbf R})$ as a vector space over {\bf R} is continuum. \end{Th} \begin{Cor} Let $G=A*B$ with $A \neq \{ 1 \}, B \neq \{ 1 \}$. If $G \neq {\mathbf Z}_2 * {\mathbf Z}_2$, then the cardinality of the dimension of $H^2_b(G;{\mathbf R})$ as a vector space over {\bf R} is continuum. \end{Cor} \noindent {\it Remark}. (1) Cor 1 is a generalization of R. Brooks' result on free groups. \noindent (2) Since $\mathbf Z_2 * \mathbf Z_2$ is amenable, $H^2_b(\mathbf Z_2 * \mathbf Z_2; \mathbf{R})$ is trivial. \begin{Cor} Let $G=A*_C B$. If $|A|=\infty$, $|C| < \infty$, and $ |B/C| \ge 2$, then the cardinality of the dimension of $H^2_b(G;\mathbf R)$ as a vector space over {\bf R} is continuum. \end{Cor} \begin{Cor} Let $G=A*_C B$. If $A$ is abelian, $|A/C| \ge 3$, and $ |B/C| \ge 2$, then the cardinality of the dimension of $H^2_b(G;\mathbf R)$ as a vector space over {\bf R} is continuum. \end{Cor} \noindent {\it Example}. $PSL_2(\mathbf Z)=\mathbf Z_2 * \mathbf Z_3$ and $SL_2(\mathbf Z)=\mathbf Z_4*_{\mathbf Z_2} \mathbf Z_6$ satisfy the assumption of Th 1. They are non-elementary word-hyperbolic groups too. For an HNN extension of a group, we show: \begin{Th} Let $G=A*_C \varphi$. If $|A/C| \ge 2, \, |A/ \varphi (C)| \ge 2$, then then the cardinality of the dimension of $H^2_b(G;\mathbf R)$ as a vector space over {\bf R} is continuum. \end{Th} \noindent {\it Remark}. In Th 1 and 2, and Cor 1, 2 and 3, we don't have to assume $G$ is finitely generated. We apply Th 1 and 2 to a group with infinitely many ends, namely, due to Stallings' structure theorem \cite{S}, we have: \begin{Th} If $G$ is a finitely generated group with infinitely many ends, then the cardinality of the dimension of $H^2_b(G;\mathbf R)$ as a vector space over {\bf R} is continuum. \end{Th} To conclude, we state a conjecture. \noindent {\it Conjecture}. Let $G$ be a group. If $H_b^2(G;{\mathbf R}) \neq \{0\}$, then $H_b^2(G;{\mathbf R})$ is infinite dimensional as a vector space over {\bf R}. \section{Quasi homomorphism and the counting function} We will review the counting function of a group w.r.t. a word. This was defined in \cite{EF} as a generalization of R. Brooks' counting function for free groups in \cite{B}. Let $G$ be a group with a (finite or infinite) set of generators and $\Gamma(G)$ the Cayley graph. For a word $w=x_1 x_2 \dots x_n$ in these generators, define $|w|=n$. Let $\overline{w}$ be the element of $G$ which is represented by the word $w$. Define $w^{-1} =x_n^{-1} \dots x_1^{-1}.$ We sometimes identify a word $w$ and the path starting at $1$ and labeled by $w$ in $\Gamma(G)$. For a path $\alpha$ labeled by $w$, define $|\alpha|=|w|$ and $\overline{\alpha}=\overline{w}$. For an element $g$ in $G$, define $|g|=\inf_{\alpha} |\alpha|$, where $\alpha$ ranges over all the paths with $\overline \alpha =g$. Let $\alpha$ be a finite path in $\Gamma(G)$. Define $|\alpha|_w$ to be the maximal number of times that $w$ can be seen as a subword of $\alpha$ without overlapping. We define $$c_w(\alpha) = \sup _{\alpha'} \{ |\alpha'|_w - ( |\alpha'|-|\overline{\alpha}|)\} = |\overline{\alpha}| - \inf_{\alpha'}(|\alpha'|-|\alpha'|_w), $$ where $\alpha'$ ranges over all the paths with the same starting point as $\alpha$ and the same finishing point. If the infimum in the definition of $c_w(\alpha)$ is attained by $\alpha'$, we say that $\alpha'$ {\it realizes} $c_w$ {\it at} $\alpha$. We have $|\overline{\alpha} | / |w| \ge c_w(\alpha) \ge 0$. If $\alpha$ is a geodesic, then $c_w(\alpha) \ge |\alpha|_w$. We define $h_w =c_w - c_{w^{-1}}$. For each $g$ in $G$, we choose $\gamma_g$ to be a path from 1 to $g$ and set $h_w(g)=h_w(\gamma_g)$. Then $h_w(g)$ does not depend on the choice of $\gamma_g$. Thus $h_w \in C^1(G;\mathbf Z)$. Let $f \in C^1(G;\mathbf R)$. If there exists a constant $D < \infty$ s.t. $$|f(gh) -f(g) -f(h) | \le D,$$ for any $g,h \in G$, then we say $f$ is a {\it quasi homomorphism} with {\it defect} $D$. Let $f$ be a quasi homomorphism with defect $D$. Then $|\delta f| \le D$ and $\delta (\delta f)=0$, thus $\delta f \in Z^2_b(G;\mathbf R)$, which defines $[\delta f] \in H^2_b(G;\mathbf R)$. Remark that we always have $[\delta f] =0$ in $H^2(G; \mathbf R)$, however, we may have $[\delta f] \not = 0$ in $H^2_b(G;\mathbf R)$ since $f$ is not necessarily in $C^1_b(G;\mathbf R)$. \section{Quasi homomorphisms of $A*_C B$} Let $G=A*_C B$ with $ |A/C| \ge 2, \, |B/C| \ge 2$. Take the set $\{A \cup B \} \backslash \{1 \}$ as a set of generators of $G$ and denote its Cayley graph by $\Gamma(G)$. Note that if a generating set is infinite, then $\Gamma$ is not locally finite. If a word $w= x_1 \dots x_n$ satisfies $n=1$ or $x_1, x_3, \dots \in A \backslash C$ (or $B \backslash C$) and $x_2, x_4, \dots \in B \backslash C$ (or $A \backslash C$, resp.), then we say $w$ is {\it reduced}. \begin{Lemma} A word $w =x_1 \dots x_n$ is reduced iff it is a geodesic in $\Gamma$. \end{Lemma} \noindent {\it Proof}. Assume $w$ is not reduced, then there exists a subword $x_ix_{i+1}$ with $x_i,x_{i+1} \in A$(or $x_i,x_{i+1} \in B$). Then $\overline{x_ix_{i+1}} \in A(\mbox{or }B \mbox{ resp.})$, thus $|\overline{x_ix_{i+1}}| \le 1$. Therefore $x_ix_{i+1}$ is not a geodesic, hence $w$ is not a geodesic. On the other hand, assume $w$ is not a geodesic. To show $w$ is not reduced by contradiction, suppose $w$ is reduced. Take a geodesic $\gamma$, hence reduced, s.t. $\overline{w} = \overline{\gamma}$. Then $|w| = |\gamma|$ since reduced words representing a same element have same length. Thus $w$ is a geodesic. This is a contradiction. We showed $w$ is not reduced. \qed \begin{Lemma} Let $w$ be a word and $\alpha$ a path. If $w^2$ is reduced, then there is a geodesic which realizes $c_w$ at $\alpha$. \end{Lemma} \noindent {\it Proof}. Since $w^2$ is reduced, $w$ is reduced. Let $\gamma$ be a path which realizes $c_w$ at $\alpha$ s.t. $|\gamma|_w$ is minimal among all the realizing paths at $\alpha$. We will show that $\gamma$ is a geodesic. If $|\gamma|_w =0$, then $\gamma$ is a geodesic. Assume $|\gamma|_w =n>0$. Write $\gamma$ as $$\gamma_1 w_1 \gamma_2 \dots w_n \gamma_{n+1},$$ where $w_i$ is a copy of $w$ and $\gamma_i$ may be an empty word. First, to show every $\gamma_i$ is reduced by contradiction, suppose some $\gamma_i$ is not reduced. Replace $\gamma_i$ by a reduced word $\gamma'_i$ with $\overline{\gamma_i}=\overline{\gamma'_i}$, then we have a new path $$\gamma' = \gamma_1 w_1 \gamma_2 \dots \gamma'_i \dots w_n \gamma_{n+1},$$ which satisfies $|\gamma'| < |\gamma|$, $|\gamma'|_w \ge |\gamma|_w$ and $\overline{\gamma}=\overline{\gamma'}$. This contradicts that $\gamma$ is a realizing path. Thus every $\gamma_i$ is reduced. Next, in order to show that $\gamma$ is reduced by contradiction, suppose not. Since $w^2$ is reduced, there is a subword $w_i \gamma _{i+1} w_{i+1}$ of $\gamma$ which is not reduced and $\gamma_{i+1}$ is not empty. Since $w_i, w_{i+1}, \gamma_{i+1}$ are reduced, one of the following four cases is occurring. \noindent (i) The last letter of $w_i$ and the initial letter of $\gamma_{i+1}$ are in $A$. \noindent (ii) The last letter of $w_i$ and the initial letter of $\gamma_{i+1}$ are in $B$. \noindent (iii) The last letter of $\gamma_{i+1}$ and the initial letter of $w_{i+1}$ are in $A$. \noindent (iv) The last letter of $\gamma_{i+1}$ and the initial letter of $w_{i+1}$ are in $B$. We treat the case (i). Write $$w_i = \dots b_1 a_1, \, \gamma_{i+1} = a_2 b_2 \dots, \, a_i \in A, b_i \in B.$$ Rewrite the subword $w_i \gamma_{i+1}$ in $\gamma$ as $$w_i \gamma_{i+1} = \dots b_1 a_1 a_2 b_2 \dots = \dots b_1 a' b_2 \dots,$$ where $a' =a_1a_2 \in A.$ This gives a new word $\gamma'$ with $|\gamma'|=|\gamma|-1,$ $|\gamma'|_w = |\gamma|_w -1$. Since $\overline{\gamma'}= \overline{\gamma}$ and $|\gamma|- |\gamma|_w = |\gamma'|-|\gamma'|_w ,$ $\gamma'$ is another realizing path with $|\gamma'|_w < |\gamma|_w $. This contradicts the choice of $\gamma$, thus $\gamma$ is reduced. By Lemma 3.1, $\gamma$ is a geodesic. A similar argument applies to the other three cases, and we omit it. \qed \begin{Lemma} Let $\alpha, \beta$ be paths starting at $1$. We have $$|c_w(\alpha)-c_w(\beta)| \le 2|\overline{\alpha^{-1} \beta}|.$$ \end{Lemma} \begin{figure} \centerline{ \epsfbox{fig1.eps} } \caption{Geodesic triangle. This illustrates Lemma 3.3.} \end{figure} \noindent {\it Proof}. Take realizing geodesics of $c_w$ at $\alpha$, $\alpha'$, and at $\beta$, $\beta'$. Then $c_w(\alpha)=|\alpha'|_w, \, c_w(\beta)=|\beta'|_w$. Take a geodesic $\sigma$ with $\overline{\alpha^{-1} \beta} =\overline{\sigma}$. Since the path $\alpha' \sigma$ satisfies $\overline{\alpha' \sigma}=\overline{\beta'}$, $$|\beta'|-|\beta'|_w \le |\alpha' \sigma|-|\alpha' \sigma|_w.$$ Since $|\alpha' \sigma| = |\alpha'| +|\sigma|$ and $|\alpha' \sigma|_w \ge |\alpha'|_w + |\sigma|_w,$ $$|\beta'|-|\beta'|_w \le |\alpha'|+ |\sigma| -|\alpha'|_w - |\sigma|_w. $$ Thus $$ c_w(\beta)=|\beta'|_w \ge |\alpha'|_w + |\beta'| -|\alpha'| -|\sigma| +|\sigma|_w \ge c_w(\alpha) -2|\sigma|,$$ since $|\alpha'|_w =c_w(\alpha), \, |\beta'| -|\alpha'| \ge - |\sigma|, \, |\sigma|_w \ge 0.$ Similarly, $c_w(\alpha) \ge c_w(\beta) - 2|\sigma|.$ Thus $|c_w(\alpha) - c_w(\beta)| \le 2|\sigma| =2|\overline{\alpha^{-1} \beta}|.$ \qed \begin{Lemma} Let $\alpha, \beta$ be paths starting at $1$. We have $$|h_w(\alpha)-h_w(\beta)| \le 4| \overline{\alpha^{-1} \beta} | .$$ \end{Lemma} \noindent {\it Proof}. $h_w =c_w -c_{w^{-1}}$ and Lemma 3.3. \qed \begin{Lemma} $$c_w(\alpha) =c_{w^{-1}}(\alpha^{-1}).$$ \end{Lemma} \noindent {\it Proof}. Clear from the definition of $c_w$. \qed \begin{Lemma} $$h_w(\alpha) = -h_w(\alpha^{-1}).$$ \end{Lemma} \noindent {\it Proof}. $h_w(\alpha^{-1})=c_w(\alpha^{-1})-c_{w^{-1}}(\alpha^{-1}) =c_{w^{-1}}(\alpha) - c_w(\alpha) = -h_w(\alpha)$. \qed \begin{Lemma} Let $\alpha$ be a reduced path. If $\alpha=\alpha_1 \alpha_2$, then $$|h_w(\alpha) - h_w(\alpha_1) - h_w(\alpha_2)| \le 10.$$ \end{Lemma} \begin{figure} \centerline{ \epsfbox{fig2.eps} } \caption{Dividing geodesics. This illustrates Lemma 3.7. $\alpha =\alpha_1 \alpha_2$, $\gamma = \gamma_1 \gamma_2$, $\sigma=\sigma_1 \sigma_2$.} \end{figure} \noindent {\it Proof}. We will show $$|c_w(\alpha) - c_w(\alpha_1) - c_w(\alpha_2)| \le 5.$$ Take realizing geodesics of $c_w$ at $\alpha_1$, $\gamma_1$, and at $\alpha_2$, $\gamma_2$. We have $|\gamma_1|_w =c_w(\alpha_1), \, |\gamma_2|_w =c_w(\alpha_2).$ Put $\gamma = \gamma_1 \gamma_2$, then $\overline{\gamma} =\overline{\alpha}$. Since $\alpha$ is reduced, it is a geodesic. Since $|\alpha|=|\alpha_1|+|\alpha_2|=|\gamma_1|+|\gamma_2|=|\gamma|$, $\gamma$ is a geodesic. Since $\gamma$ is a geodesic with $\overline{\gamma}=\overline{\alpha}$, $c_w(\alpha) \ge |\gamma|_w$. Thus $$c_w(\alpha) \ge |\gamma|_w \ge |\gamma_1|_w + |\gamma_2|_w = c_w(\alpha_1) + c_w(\alpha_2).$$ On the other hand, take a realizing geodesic $\sigma$ at $\alpha$, then $c_w(\alpha)=|\sigma|_w$. Since $\alpha$ and $\sigma$ are reduced, there exists subdivision of $\sigma$, $\sigma =\sigma_1 \sigma_2$ s.t. $\overline{\alpha_1^{-1} \sigma_1} = \overline{\alpha_2 \sigma_2^{-1}} =c, $ for some $c \in C$. Since $|c| \le 1$, from Lemma 3.3 and 3.5. $$|c_w(\alpha_i)-c_w(\sigma_i)| \le 2, \, i=1,2.$$ Since $\sigma_1, \sigma_2$ are geodesics, $|\sigma_i|_w \le c_w(\sigma_i), \, i=1,2.$ Thus $$c_w(\alpha)= |\sigma|_w \le |\sigma_1|_w + |\sigma_2|_w +1 \le c_w(\sigma_1) + c_w(\sigma_2) +1 \le c_w(\alpha_1) + c_w(\alpha_2) +5.$$ We showed $|c_w(\alpha) - c_w(\alpha_1) -c_w(\alpha_2)| \le 5$. $|c_{w^{-1}}(\alpha) - c_{w^{-1}}(\alpha_1) -c_{w^{-1}}(\alpha_2)| \le 5$ is similar. Since $h_w=c_w -c_{w^{-1}}$, we get $|h_w(\alpha) - h_w(\alpha_1) - h_w(\alpha_2)| \le 10.$ \qed \begin{Prop} Let $w$ be a word. We have $$|\delta h_w| \le 78. $$ \end{Prop} \begin{figure} \centerline{ \epsfbox{fig3.eps} } \caption{Dividing geodesic triangle. This illustrates Prop 1. $c_1, c_2, c_3 \in C$.} \end{figure} \noindent {\it Proof}. Let $x,y$ be elements in $G$. $\delta h_w(x,y) =h_w(x) + h_w(y) -h_w(xy).$ We will show $|h_w(xy)-h_w(x)-h_w(y)| \le 78.$ Take geodesics, hence reduced paths, $\alpha, \beta$ and $\gamma$ with $\overline{\alpha} =x, \overline{\beta} =y, \overline{\gamma} =xy$. Since $\overline{\alpha \beta} = \overline{\gamma}$ and $\alpha, \beta, \gamma$ are reduced, there exist subdivisions of $\alpha, \beta, \gamma$, $$\alpha= \alpha_1 \alpha_2 \alpha_3, \, \beta =\beta_1 \beta_2 \beta_3, \, \gamma =\gamma_1 \gamma_2 \gamma_3, $$ s.t. $$\overline{\gamma_1^{-1} \alpha_1} =c_1, \, \overline{\alpha_3 \beta_1} =c_2, \, \overline{\beta_3 \gamma_3^{-1}} =c_3, $$ for some $c_1, c_2, c_3 \in C$ and $\overline{\alpha_2}, \overline{\beta_2}, \overline{\gamma_2}$ are (simultaneously) in $A$ or $B$. Since $\overline{\alpha_2}$ is in $A$ or $B$, $c_w(\alpha_2), c_{w^{-1}}(\alpha_2) \le |\overline{\alpha_2}| \le 1$. Thus $|h_w(\alpha_2)| \le 2$. Similarly, $|h_w(\beta_2)|, |h_w(\gamma_2)| \le 2$. Using Lemma 3.7 twice for $\alpha = ((\alpha_1 \alpha_2) \alpha_3)$, $$|h_w(\alpha) -h_w(\alpha_1) -h_w(\alpha_2)-h_w(\alpha_3)| \le 20, $$ thus $|h_w(\alpha) -h_w(\alpha_1) -h_w(\alpha_3) | \le 22.$ Similarly, $$|h_w(\beta) - h_w(\beta_1) - h_w(\beta_3)| \le 22, $$ $$|h_w(\gamma) - h_w(\gamma_1) - h_w(\gamma_3)| \le 22. $$ Thus \begin{eqnarray*} && |h_w(\alpha) + h_w(\beta) -h_w(\gamma)| \\ & \le & |h_w(\alpha_1) +h_w(\alpha_3) + h_w(\beta_1) + h_w(\beta_3) -h_w(\gamma_1) - h_w(\gamma_3) | + 66\\ & \le & |h_w(\alpha_1) - h_w(\gamma_1)| + | h_w(\beta_1) - h_w(\alpha_3^{-1})| + |h_w(\gamma_3^{-1}) - h_w(\beta_3^{-1})| + 66 \\ &\le & 4(|c_1| +|c_2| +|c_3|) + 66 \le 78, \end{eqnarray*} since $|c_i| \le 1$ and Lemma 3.4 and 3.6. By definition, $h_w(\alpha)=h_w(x), h_w(\beta)=h_w(y), h_w(\gamma)=h_w(xy)$, thus $$|h_w(x) + h_w(y) -h_w(xy) | \le 78.$$ \qed \section{Choice of words for $A*_C B$} The goal of this section is to prove the following proposition. \begin{Prop} Let $G=A*_CB$ with $|C \backslash A /C| \ge 3$ and $|B/C| \ge 2$. Then there exist words $w_i, \, 0 \le i < \infty$ which satisfy the following conditions. \noindent (1) For any $i \ge 0$ and any $n \ge 1$, $h_{w_i}(w_i^n) =n$. \noindent (2) For any $j > i \ge 0$ and any $n \ge 1$, $h_{w_j}(w_i^n) =0$. \noindent (3) For any $i \ge 0$, $\overline{w_i} \in [G,G]$. \end{Prop} Let $G=A*_CB$ with $|C \backslash A /C| \ge 3$ and $|B/C| \ge 2$. Let $w$ and $w'$ be reduced paths starting at a same point and finishing at a same point, $$w=a_1 b_1 \dots a_n b_n, $$ $$w'=a_1' b_1' \dots a_m' b_m',$$ where $a_1, b_n, a_1', b_m'$ may be empty. Since $\overline{w}=\overline{w'}$ and $w$ and $w'$ are reduced, $n=m$ and there exist $c_1, \dots c_n$ and $c_0', c'_1, \dots c'_n$ in $C$ s.t. $$c_{i-1}' a_i c_i^{-1} = a_i', \, c_ib_i c_i'^{-1} = b_i',$$ for $1\le ^{\forall}i \le n$, where $c_0'=c_n'=1$. See Figure 4. \begin{figure} \centerline{\epsfxsize=12cm \epsfbox{fig4.eps} } \caption{Reduced paths $w, w'$ with common end points. $\overline{w} = \overline{w'}$.} \end{figure} Let $v$ be a subword of $w$, $$v=a_ib_i \dots a_j b_j,$$ where $a_i, b_j$ may be empty. Define a subword of $w'$, denoted by $P(v)$, by $$P(v) =a_i' b_i' \dots a_j' b_j'.$$ \begin{figure} \centerline{\epsfxsize=12cm \epsfbox{fig5.eps} } \caption{Definition of $P(v)$.} \end{figure} Let $v'$ be a subword of $w'$. If $v'$ is a subword of $P(v)$, we say $v$ {\it covers} $v'$. If $v'=P(v)$, then we say $v$ {\it faces} $v'$. Since $|C \backslash A / C| \ge 3$, we can choose $a_1, a_2 \in A$ s.t. $a_1, a_2 \in A \backslash C,$ and $a_2 \not \in Ca_1C$. Take $ b \in B \backslash C$. Define words $w_i, 0 \le i < \infty$ by \begin{eqnarray} w_i &=& (a_1b)^{10^i} (a_1^{-1}b^{-1})^{10^i} (a_2b)^{10^i} (a_2^{-1}b^{-1})^{10^i} \label{words} \nonumber \\ && (a_1b)^{4 \cdot 10^i} (a_1^{-1}b^{-1})^{4 \cdot 10^i} (a_2b)^{4 \cdot 10^i} (a_2^{-1}b^{-1})^{4 \cdot 10^i}. \nonumber \end{eqnarray} We write the subword $(a_1b)^{4 \cdot 10^i}$ by $w_i(1,+)$ and the subword $(a_1^{-1}b^{-1})^{4 \cdot 10^i}$ by $w_i(1,-)$. \begin{Lemma} The words $w_i, 0 \le i < \infty$ satisfy the following conditions. \noindent (1) For any $i \ge 0$ and any $n \ge 1$, $w_i^n$ is reduced. \noindent (2) For any $i \ge 0$, $|w_i|=40 \cdot 10^i$ and $|w_i(1,\pm)| =8 \cdot 10^i.$ \noindent (3) For any $i \ge 0$, $\overline{w_i} \in [G,G].$ \end{Lemma} \noindent {\it Proof}. (1) is clear from the choice of $a_1, a_2$, and $b$. (2) is obvious. For (3), use $ \overline{(a_ib)^p(a_i^{-1}b^{-1})^p} \in [G,G]$ for $i=1,2$ and any $p \ge 1$. \qed \begin{Lemma} In any pair of paths having a same starting point and a same finishing point, the following conditions hold. \noindent (1) $a_1$ cannot face $a_2$. \noindent (2) For any $i \ge 0$, $w_i^2$ cannot cover $w_i^{-1}$. \noindent (3) For any $i < j$, $w_j(1,+)$ cannot cover $w_i$ and $w_j(1,-)^{-1}$ cannot cover $w_i$. \noindent (4) For any $k>0$ and any $i 0$. Then $w_j(1,+)$ covers some $w_i$ of $w_i^k$ since $2|w_i| \le |w_j(1,+)|$, which contradicts (3). Assume $w_i^k$ covers $w_j^{-1}$ for some $i 0$. Then $w_j(1,-)^{-1}$ covers some $w_i$ of $w_i^k$, which contradicts (3). \qed \begin{Lemma} $c_{w_i}$ and $c_{w_i^{-1}}$ satisfy the following conditions. \noindent (1) For any $n \ge 1$ and any $i \ge 0$, $c_{w_i} (w_i^n)=n.$ \noindent (2) For any $n \ge 1$ and any $i \ge 0$, $c_{w_i^{-1}}(w_i^n) =0.$ \noindent (3) For any $j > i \ge0$ and any $n \ge 1$, $c_{w_j}(w_i^n)=0.$ \noindent (4) For any $j > i \ge0$ and any $n \ge 1$, $c_{w_j^{-1}}(w_i^n) =0.$ \end{Lemma} \noindent {\it Proof}. (1) Since $w_i^n$ is reduced, $w_i^n$ is a geodesic. Thus $c_{w_i}(w_i^n) \ge |w_i^n|_{w_i} =n$. On the other hand, $c_{w_i}(w_i^n) \le |\overline{w_i^n}|/|w_i| =n$. Thus $c_{w_i}(w_i^n) =n.$ \noindent (2) To show $c_{w_i^{-1}}(w_i^n) =0$ by contradiction, assume $c_{w_i^{-1}}(w_i^n) > 0$. Take a realizing geodesic of $c_{w_i^{-1}}$ at $w_i^n$, $\alpha$. Then $|\alpha|_{w_i^{-1}} > 0$. Fix a subword labeled by $w_i^{-1}$ in $\alpha$. Since $w_i^n$ and $\alpha$ are reduced, there is a subword of $w_i^n$ labeled by $w_i^2$ which covers the $w_i^{-1}$ in $\alpha$. But $w_i^2$ cannot cover $w_i^{-1}$ by Lemma 4.2(2), a contradiction. See Figure 7. \begin{figure} \centerline{ \epsfbox{fig7.eps} } \caption{$w_i^2$ cannot cover $w_i^{-1}$.} \end{figure} \noindent (3) To show the claim by contradiction, assume $c_{w_j}(w_i^n) > 0$ for some $i < j$ and $n$. Take a realizing geodesic $\alpha$ of $c_{w_j}$ at $w_i^n$. Then $|\alpha|_{w_j} >0$. Fix some subword labeled by $w_j$ in $\alpha$. Since $\alpha$ and $w_i^n$ are reduced, there exists a subword labeled by $w_i^k$ in $w_i^n$ which covers the $w_j$ in $\alpha$. But $w_i^k$ cannot cover $w_j$ by Lemma 4.2(4), a contradiction. \noindent (4) Similar to (3). \qed Now we are in a position to prove Proposition 2. \noindent {\it Proof of Proposition 2}. Let $w_i, 0 \le i < \infty$ be the set of words defined before in this section. We will show they are the words we want. Since $h_{w_i} = c_{w_i} -c_{w_i^{-1}}$, $h_{w_i}(w_i^n)=n$ by Lemma 4.3(1) and (2). By Lemma 4.3(3) and (4), $h_{w_j}(w_i^n)=0$ for $i$ with $|A/C| \ge 2, \, |A/ \varphi (C)| \ge 2$. Let $g$ be $$g =a_1 t^{n_1} a_2 t^{n_2} \dots a_I t^{n_I} a_{I+1}, $$ with $a_i \in A$ and $ n_i \neq 0$, where $a_1, a_{I+1}$ may be empty. Suppose $g$ satisfies the following two conditions (i) and (ii) for $1 \le ^{\forall} i \le I-1$. \noindent (i) If $n_i >0, n_{i+1} <0$, then $a_{i+1}\notin C$. \noindent (ii) If $n_i < 0, n_{i+1} > 0$, then $a_{i+1} \notin \varphi(C)$. \noindent Then we say $g$ is {\it reduced}. The following fact is known as Britton's lemma. \begin{Lemma} [{\bf Britton, \cite{LS}}] Suppose $1 \le I$. If $g$ is reduced, then $g \neq 1$ in $G$. \end{Lemma} As an application of Britton's lemma, we have: \begin{Lemma} Let $$g=a_1 t^{n_1} a_2 t^{n_2} \dots a_I t^{n_I} a_{I+1}, \, \, a_i \in A$$ $$h=b_1 t^{m_1} b_2 t^{m_2} \dots b_J t^{m_J} b_{J+1}, \, \, b_j \in A$$ be reduced with $n_i = \pm 1, \, m_j = \pm 1.$ If $g=h$ in $G$, then $$I=J, \, n_1=m_1, \dots ,n_I=m_I,$$ and $$a_{I+1}b_{I+1}^{-1}, \, \, t^{n_I}a_{I+1}b_{I+1}^{-1}t^{-m_I}, \, \, a_It^{n_I}a_{I+1}b_{I+1}^{-1}t^{-m_I}b_I^{-1}, \dots $$ $$\dots , t^{n_1}a_2t^{n_2}a_3 \dots a_{I+1}b_{I+1}^{-1} \dots b_3 ^{-1}t^{-m_2}b_2^{-1}t^{-m_1}$$ are in $C$ or $\varphi(C)$, especially, in $A$. \end{Lemma} \noindent {\it Proof}. W.l.o.g., we assume $J \le I$. $h^{-1}= b_{J+1}^{-1}t^{-m_J}b^{-1}_J t^{-m_{J-1}} \dots b^{-1}_1$ is reduced since $h$ is reduced. Put $$d_{I+1} =a_{I+1} b_{J+1}^{-1}, $$ then $d_{I+1} \in A$ and $$gh^{-1} = (a_1 t^{n_1} a_2 t^{n_2} \dots a_I t^{n_I}) d_{I+1} (t^{-m_J}b^{-1}_J t^{-m_{J-1}} \dots b^{-1}_1)$$ in $G$. Since $gh^{-1} =1$ in $G$, the expression on the right hand side is not reduced by Britton's lemma. But, since $a_1 t^{n_1} a_2 t^{n_2} \dots a_I t^{n_I}$ and $t^{-m_J}b^{-1}_J t^{-m_{J-1}} \dots b^{-1}_1$ are reduced, we have $$n_I =m_J=1 (\mbox{or} -1), \, \, d_{I+1} \in C (\varphi(C), \mbox{resp.}),$$ and $t^{n_I} d_{I+1} t^{-m_J} \in \varphi(C)$($C$, resp.). Put $$c_I = t^{n_I} d_{I+1} t^{-m_J}, \, \, d_I = a_I c_I b_J^{-1}.$$ Clearly $d_I \in A$ and $$gh^{-1} = (a_1 t^{n_1} a_2 t^{n_2} \dots a_{I-1} t^{n_{I-1}}) d_I (t^{-m_{J-1}}b^{-1}_{J-1} t^{-m_{J-2}} \dots b^{-1}_1)$$ in $G$. By Britton's lemma, the expression on the right hand side is not reduced. Thus $$n_{I-1}=m_{J-1}=1(\mbox{or} -1), $$ $$d_I \in C (\varphi(C), \mbox{resp}.), \, \, \mbox{and } c_{I-1}=t^{n_{I-1}} d_I t^{-m_{J-1}} \in \varphi(C) (C, \mbox{resp}.).$$ We define $d_{I-1}, d_{I-2}, \dots, d_{I-J+2}$ and $c_{I-1}, c_{I-2}, \dots, c_{I-J+1}$ inductively by $$c_i=t^{n_i} d_{i+1} t^{-m_{J-I+i}}, \, \, \, d_i = a_i c_i b^{-1}_{J-I+i}.$$ See Figure 8. \begin{figure} \centerline{\epsfxsize=12cm \epsfbox{fig8.eps} } \caption{This illustrates Lemma 6.2. $d_i, c_i \in A$ for $I-J+1 \le ^{\forall} i \le I$. If $J < I$, then it contradicts Britton's lemma.} \end{figure} Using Britton's lemma repeatedly, we have $$n_{I-J+j}=m_j, \, 1 \le ^{\forall} j \le J$$ and $$d_{I+1}, d_I, \dots, d_{I-J+2}, c_I, c_{I-1}, \dots, c_{I-J+1} \in C \mbox{ or } \varphi(C).$$ Put $$d_{I-J+1} = a_{I-J+1} c_{I-J+1} b_1^{-1}.$$ Clearly $d_{I-J+1} \in A$. To complete the proof, it suffices to show $J=I$. In order to show it by contradiction, suppose $J < I$. Then $$gh^{-1} = a_1t^{n_1}a_2 \dots t^{n_{I-J-1}} a_{I-J}t^{n_{I-J}} d_{I-J+1} $$ in $G$ and the expression on the right hand side is reduced, which contradicts Britton's lemma, since $gh^{-1}=1$ in $G$. We got $I=J$. \qed We take $\{ t\} \cup A \backslash \{ 1 \}$ as a set of generators of $G$ and write the Cayley graph of $G$ w.r.t. this set by $\Gamma$. \begin{Lemma} If a path $\alpha$ is a geodesic in $\Gamma$, then it is reduced. \end{Lemma} \noindent {\it Proof}. If a path $\alpha$ is not reduced, then we can make it shorter using $tct^{-1} = \varphi(c)$, hance $\alpha$ is not a geodesic. \qed \noindent {\it Remark}. Compare Lemma 6.3 with Lemma 3.1. A reduced path is not always a geodesic in this case. For example, the left hand side of $\varphi(c)^{-1} t c =t$ for $c \in C \backslash \{1\}$ is reduced but not a geodesic. \begin{Lemma} Let $\alpha$ be a path and $w$ a word. If $w^2$ is reduced, then there is a reduced path $\beta$ which realizes $c_w$ at $\alpha$. \end{Lemma} \noindent {\it Proof}. Take a path $\beta$ which realizes $c_w$ at $\alpha$ s.t. $|\beta|_w$ is minimal among all the realizing paths. Then a similar argument to the proof of Lemma 3.2 shows $\beta$ is reduced. We omit the detail. \qed \begin{Lemma} Let $\alpha, \beta$ be paths starting at $1$. We have $$|c_w(\alpha)-c_w(\beta)| \le 2|\overline{\alpha^{-1} \beta}|.$$ $$|h_w(\alpha)-h_w(\beta)| \le 4|\overline{\alpha^{-1} \beta}|.$$ \end{Lemma} \noindent {\it Proof}. Similar to Lemma 3.3 and 3.4. \qed \begin{Lemma} Let $\alpha$ be a reduced path. If $\alpha= \alpha_1 \alpha_2$, then $$|h_w(\alpha) - h_w(\alpha_1) -h_w(\alpha_2) | \le 10.$$ \end{Lemma} \noindent {\it Proof}. Similar to Lemma 3.7. \qed \begin{Prop} $$|\delta h_w | \le 78.$$ \end{Prop} \noindent {\it Proof}. The outline is similar to the proof of Proposition 1. Let $x,y$ be elements in $G$. We show $|h_w(xy)-h_w(x)-h_w(y)| \le 78.$ Take reduced paths $\alpha, \beta, \gamma$ with $\overline{\alpha}=x, \overline{\beta}=y, \overline{\gamma}=xy$. Since $\overline{\alpha \beta} = \overline{\gamma}$ and $\alpha, \beta, \gamma$ are reduced, by Lemma 6.1 and 6.2, there exist subdivisions of $\alpha, \beta, \gamma$, $$\alpha=\alpha_1 \alpha_2 \alpha_3, \, \beta =\beta_1 \beta_2 \beta_3, \, \gamma=\gamma_1 \gamma_2 \gamma_3, $$ s.t. $\overline{\gamma^{-1}_1 \alpha_1}= c_1, \overline{\alpha_3 \beta_1}=c_2, \overline{\beta_3 \gamma_3^{-1}}=c_3$, for some $c_1, c_2, c_3 \in C \cup \varphi(C),$ and $\overline{\alpha_2}, \overline{\beta_2}, \overline{\gamma_2} \in A$. By Lemma 6.6, $$|h_w(\alpha) -h_w(\alpha_1) -h_w(\alpha_2) -h_w(\alpha_3) | \le 20.$$ Since $|h_w(\alpha_2) | \le 2,$ $|h_w(\alpha) -h_w(\alpha_1) - h_w(\alpha_3) | \le 22.$ Similarly, $$|h_w(\beta) -h_w(\beta_1) - h_w(\beta_3) | \le 22, \, |h_w(\gamma) -h_w(\gamma_1) - h_w(\gamma_3) | \le 22.$$ By a similar argument to the proof of Prop 1, $$|h_w(x)+h_w(y)-h_w(xy)| = |h_w(\alpha) + h_w(\beta) -h_w(\gamma)| \le 78.$$ \qed \section{Choice of words for $A*_C\varphi$} Let $G=A*_C \varphi$ with $|A/C| \ge 2, \, |A/ \varphi (C)| \ge 2$. \begin{Lemma} Let $w$ be $$w=t^{n_1} a_1 t^{n_2} a_2 \dots t^{n_I} a_I,$$ s.t. $a_i \in A \backslash \{ 1 \}, n_i \in {\mathbf Z} \backslash \{0\}$ for $1 \le ^{\forall} i \le I$. We denote the set of the following conditions (1.1), \dots, (1.4) by Condition I and (2.1), \dots, (2.4) by Condition II. \noindent $(1.1) \, 0 < n_1, n_3, n_5, \dots $. \noindent $(1.2) \,0 > n_2, n_4, n_6, \dots$. \noindent $(1.3) \, a_1, a_3, a_5, \dots \notin C$. \noindent $(1.4) \, a_2, a_4, a_6, \dots \notin \varphi(C)$. \noindent $(2.1) \, 0 > n_1, n_3, n_5, \dots $. \noindent $(2.2) \, 0 < n_2, n_4, n_6, \dots$. \noindent $(2.3) \, a_1, a_3, a_5, \dots \notin \varphi(C)$. \noindent $(2.4) \,a_2, a_4, a_6, \dots \notin C$. If either the Condition I or II holds, then $w$ is a geodesic in $\Gamma$. \end{Lemma} \noindent {\it Proof}. We treat the case that the Condition I holds. The other case is similar. Under this condition, clearly $w$ is reduced. Let $\gamma$ be a geodesic with $\overline{w} = \overline{\gamma}$. Then by Lemma 6.3, $\gamma$ is reduced. By Lemma 6.2, we have $$\gamma =b_0 \tau_1 b_1 \tau_2 b_2 \dots \tau_I b_I,$$ s.t. $b_i \in A \backslash \{ 1 \}$ for $1 \le i \le I-1$, and $b_0, b_I \in A \backslash \{ 1 \}$ or is empty, and \begin{eqnarray*} \tau_i & = & t b_{i,1} t b_{i,2} \dots t b_{i,n_i -2} t b_{i,n_i-1} t, \mbox{ if $i$ is odd}, \\ \tau_i & = & t^{-1} b_{i,1} t^{-1} b_{i,2} \dots t^{-1} b_{i,n_i -2} t^{-1} b_{i,n_i-1} t^{-1}, \mbox{ if $i$ is even}, \end{eqnarray*} where $b_{i,j} \in A \backslash \{ 1 \}$ or is empty, for $1 \le ^{\forall} i \le I$ and $1 \le ^{\forall} j \le n_i -1$. We will show $b_I$ is not empty. To show this by contradiction, assume $b_I$ is empty. Then \begin{eqnarray*} w \gamma^{-1} & = & \dots t a_I t^{-1} \dots, \mbox{ if $I$ is odd}, \\ w \gamma^{-1} & = & \dots t^{-1} a_I t \dots, \mbox{ if $I$ is even.} \end{eqnarray*} Since $w$ and $\gamma^{-1}$ are reduced, $w \gamma^{-1}$ is reduced by (1.3) if $I$ is odd or by (1.4) if $I$ is even. Then it follows from Britton's lemma that $\overline{w \gamma^{-1}} \neq 1$, a contradiction. We got $b_I$ is not empty. Thus $$|\gamma| \ge \sum_{i=1}^I |\tau_i| + I \ge \sum_{i=1}^I |n_i| +I = |w|.$$ We have the first inequality because $b_1, \dots, b_I$ are not empty, and the second one since the number of $t$'s in $\tau_i$ is $n_i$ for $1 \le ^{\forall} i \le I$. Since $\gamma$ is a geodesic, we have $|\gamma| = |w|$, hence $w$ is a geodesic. \qed \begin{Lemma} Take $g \in A \backslash C$ and $h \in A \backslash \varphi(C)$ and fix them. \noindent Let $w_i, 0 \le i < \infty$, be words s.t. $$w_i = t^{10^i} g t^{-10^i} h t^{10^i} g^{-1} t^{-10^i} h^{-1} t^{2 \cdot 10^i} g t^{-2 \cdot 10^i} h t^{3 \cdot 10^i} g^{-1} t^{-3 \cdot 10^i} h^{-1}.$$ Then the words satisfy the following conditions. \noindent (1) For any $i \ge 0$ and any $n \ge 1$, $c_{w_i} (w_i^n) =n.$ \noindent (2) For any $i \ge 0$ and any $n \ge 1$, $c_{w_i^{-1}} (w_i^n) =0.$ \noindent (3) For any $j > i \ge 0$ and any $n \ge 1$, $c_{w_j^{\pm 1}} (w_i^n) =0.$ \noindent (4) For any $i \ge 0$, $\overline{w_i} \in [G,G]$. \end{Lemma} \noindent {\it Proof}. (1) By Lemma 7.1, $w_i^n$ is a geodesic. Thus $c_{w_i} (w_i^n) =n$. \noindent (2) In order to show $c_{w_0^{-1}}(w_0^n) =0$ for any $n \ge 1$ by contradiction, suppose $c_{w_0^{-1}}(w_0^n) > 0$ for some $n$. By Lemma 6.4, take a reduced path $\alpha$ s.t. $\overline{\alpha} = \overline{w_0^n}$ and $|\alpha|_{w_0^{-1}} > 0$. We describe the order of $t$'s and $t^{-1}$'s appearing in $w_0$ by $W_0$, $$W_0 = +\, -\, +\, -\, +\, +\, -\, -\, +\, +\, +\, -\, -\, -,$$ where $+$ stands for $t$, $-$ for $t^{-1}$ and we ignore $g^{\pm 1}, h^{\pm 1}$ in $w_0$. Then $W_0^n$ represents $w_0^n$. Let $\Lambda $ be the order of $t$'s and $t^{-1}$'s in $\alpha$. Applying Lemma 6.2 for $\alpha$ and $w_0^n$, we get $W_0^n= \Lambda$. $\Lambda$ contains $W_0^{-1}$ as a subpiece since $|\alpha|_{w_0^{-1}} > 0$, where $$W_0^{-1} = +\, +\, +\, -\, -\, -\, +\, +\, -\, -\, +\, -\, +\, -\, .$$ But it is easy to check that $W_0^n$ cannot contain $W_0^{-1}$ as a subset, thus we get a contradiction. Hence $c_{w_0^{-1}}(w_0^n) =0$. Similarly, we can show that $c_{w_i^{-1}}(w_i^n) =0$ for any $i \ge 1$ and any $n \ge 1$. \noindent (3) Let $W_i, \, 1 \le i < \infty$ be the order of $t$'s and $t^{-1}$'s in $w_i$. To show $c_{w_j}(w_i^n) =0$ for all $n \ge 1$ and all $j > i >0$ by contradiction, assume $c_{w_j}(w_i^n) > 0$ for some $n \ge 1$ and some $j > i >0$. Then, by our assumption, $W_j$ must be contained in $W_i^n$ as a subword. This is a contradiction since $W_j$ has $3 \cdot 10^j$ consecutive $t$'s, but $W_i$ has at most $3 \cdot 10^i$ consecutive $t$'s. Thus we have $c_{w_j}(w_i^n) =0$. $c_{w_j^{-1}}(w_i^n) =0$ is similar. \noindent (4) Set $T=t^{10^i}$, then $$w_i=[T,g][gh,[T,g^{-1}]\, ][T,g^{-1}][g,h][T^2,g][gh,[T^3,g^{-1}]\, ] [T^3,g^{-1}][g,h],$$ thus $\overline{w_i} \in [G,G]$. \qed \begin{Prop} Let $G=A*_C \varphi$ with $|A/C| \ge 2, \, |A / \varphi(C)| \ge 2$. There exist words $w_i, \, 0 \le i < \infty$ which satisfy the following conditions. \noindent (1) For any $i \ge 0$ and any $n \ge 1$, $h_{w_i} (w_i^n) =n. $ \noindent (2) For any $j > i \ge 0$ and any $n \ge 1$, $h_{w_j} (w_i^n) =0.$ \noindent (3) For any $i \ge 0$, $\overline{w_i} \in [G,G]$. \end{Prop} \noindent {\it Proof}. The words $w_i, \, 0 \le i < \infty$ in Lemma 7.2 clearly satisfy (1), (2) and (3). \qed \section{Proofs of Th 2 and Th 3} \noindent {\it Proof of Th 2}. We can show Th 2 using Prop 3 and Prop 4 as we showed Th 1 using Prop 1 and Prop 2. We omit the detail. \qed \noindent {\it Proof of Th 3}. By J. Stallings' structure theorem \cite{S}, $G$ is either \noindent (1) $A*_C B$ with $|C| < \infty, \, |A/C| \ge 3$ and $|B/C| \ge 2$, \noindent or \noindent (2) $A *_C \varphi$ with $|C| < \infty$ and $|A/C| \ge 2$. In the case (1), if $|A| = \infty$ or $|B|=\infty$, then we have the conclusion by Cor 2. If $|A| < \infty$ and $|B| < \infty$, then $G$ is word-hyperbolic. Since $G$ has infinitely many ends, it is non-elementary. It is known that the conclusion of Th 3 holds for a non-elementary word-hyperbolic group, \cite{EF}. In the case (2), since $|A/C| \ge 2$ and $|C| < \infty$, we have $|A/ \varphi (C)| \ge 2$. Apply Th 2. \qed \begin{thebibliography}{BaGh3} \bibitem[BaGh1]{BaGh1} J. Barge, E. Ghys, Surfaces et cohomologie born\'{e}e, Invent. Math., 92, 1988, 509-526. \bibitem[BaGh2]{BaGh2} J. Barge, E. Ghys, Cocycles born\'{e}s et actions de groupes sur les arbres r\'{e}els, in ``Group Theory from a Geometric Viewpoint'', World Sci. Pub., New Jersey, 1991, 617-622. \bibitem[BaGh3]{BaGh3} J. Barge, E. Ghys, Cocycles d'Euler et de Maslov, Math. Ann., 294, no 2, 1992, 235-265. \bibitem[Bav]{Bav} C. Bavard, Longueur stable des commutateurs, L'Enseignement Math., 37, 1991, 109-150. \bibitem[B]{B} R. Brooks, Some remarks on bounded cohomology, Ann. Math. 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