Sato, Takamichi

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Title Schreier coset graphs of Baumslag-Solitar groups (Advances inGeneral Topology and their Problems)Author(s)Sato, TakamichiCitation数理解析研究所講究録 = RIMS Kokyuroku (2019), 2110:21-27Issue Date2019-04URL http://hdl.handle.net/2433/251960RightType Departmental Bulletin PaperTextversionpublisherKyoto University21Schreier coset graphs of Baumslag‐Solitar groupsTakamichi SatoGraduate School of Fundamental Science and Engineering, Waseda University1. IntroductionThis article is an extended version of the talk given at the RIMS Meeting on SetTheoretic and Geometric Topology held in Kyoto University from June 20 to June 22,2018.Let m and  n be non‐zero integers. The group which has the presentation \{A,   B|AB^{m}=B^{n}A\rangle is called the Baumslag‐Solitar group and denoted by  BS(m, n) .In 1962, G. Baumslag and D. Solitar [1] introduced these groups and showed that BS(3,2) is the first example of non‐Hopfian groups with one defining relation. Sincethen these groups have served as a proving ground for many new ideas in combinatorialand geometric group theory (see for example, [2, 3]).Schreier coset graphs are generalizations of the Cayley graph of a group  G , whichare constructed for each choice of a subgroup of  G and a generating set of  G . In general,to construct the Cayley graph of a group or Schreier coset graphs is difficult. Howeveronce we have the appropriate Cayley or Schreier graphs, we can use them as discretemodels and may learn, from combinatorial and geometric viewpoints, some propertiesof the original group or its subgroups. Recently, in [5, 6], D. Savchuk constructedSchreier graphs of Thompson’s group  F from a motivation to study amenability ofthe group.In this article, we focus on the group  BS(1, n) , where  n\geq 2 , which has an actionon the real line, and we explicitly construct Schreier graphs of the group for stabilizersof all points under this action. As its consequence, we classify the Schreier graphs upto isomorphism. This leads to a relevance to presentations for the stabilizers whichturn out to be infinite index subgroups in the group  BS(1, n) .2. Definitions and notationsLet  m and  n be nonzero integers. The group which has the presentation \{A,   B|AB^{m}=B^{n}A\rangle is called the Baumslag‐Solitar group and it is denoted by BS(m, n) . In the case of  m=1 and  n\geq 2 the group  BS(1, n) has a geometricrepresentation. That is, we define two affine maps  a and  b of the real line  \mathbb{R} by22TAKAMICHI SATO a(x)=nx and  b(x)=x+1 respectively. Then  BS(1, n) is isomorphic to the group \{\{a, b\}\rangle . We note that \{\{a, b\}\rangle=\{g:\mathbb{R}arrow \mathbb{R}|g(x)=n^{i}x+j/n^{k}, i, j, k\in \mathbb{Z}\}.A labelled directed graph denoted by  (V, E, L, \alpha, \beta, l) consists of a nonempty set V of vertices, a set  E of edges, a set  L of labels and three mappings  \alpha :  Earrow V, \beta :  Earrow V , and  l :  Earrow L . The vertices  \alpha(e) and  \beta(e) are called the initial and theterminal vertices of the edge  e , respectively.A marked labelled directed graph denoted by  (V, E, L, \alpha, \beta, l, v_{0}) is a labelled di‐rected graph with a distinguished vertex  v_{0} called the marked vertex.For  i\in\{1,2\} let  \Gamma_{i} be labelled directed graph  (V_{i}, E_{i}, L_{i}, \alpha_{i}, \beta_{i}, l_{i}) .  \Gamma_{1} is said tobe isomorphic to  \Gamma_{2} if there exist bijections  f :  V_{1}arrow V_{2},  \psi :  E_{1}arrow E_{2} , and  \gamma :  L_{1}arrow L_{2}such that  \alpha_{2}(\psi(e))=f(\alpha_{1}(e)),  \beta_{2}(\psi(e))=f(\beta_{1}(e)) , and  l_{2}(\psi(e))=\gamma(l_{1}(e)) for all e\in E_{1} . In particular, if  L_{1}=L_{2}=L and  \gamma=1,  \Gamma_{1} is said to be  L ‐isomorphic to  \Gamma_{2}.For  i\in\{1,2\} let  \Gamma_{i} be marked labelled directed graph.  \Gamma_{1} is said to be isomorphicto  \Gamma_{2} if  \Gamma_{1} is isomorphic to  \Gamma_{2} as labelled directed graphs and the mapping betweenvertices preserves the marked vertices.Let  S be a generating set of a group  G . The generating set  S is symmetric if S=S^{-1}.DEFINITION 1. Let  G be a finitely generated group,  S be a symmetric finitegenerating set of  G and  M be a set. Let  \varphi :  Garrow Aut(M) be a homomorphism, where Aut(M) is the set of all bijections of  M onto itself. The Schreier graph denoted by (M, S, \varphi) is a labelled directed graph  (M, M\cross S, S, \alpha, \beta, l) such that  \alpha(m, s)=m, l(m, s)=s , and  \beta(m, s)=\varphi(s)(m) . The Schreier graph with a marked vertex denotedby  (M, S, \varphi, m_{0}) is a Schreier graph with a marked vertex  m_{0}\in M.Let  H be a subgroup of a group  G with a symmetric finite generating set  S and G/H be the set of all left cosets of  H in  G . The Schreier coset graph denoted by (G/H, S) is a Schreier graph  (G/H, S, \varphi_{0}) where  \varphi_{0} :  Garrow Aut(G/H) is defined by \varphi_{0}(x)(gH)=xgH.The next proposition will help us to describe Schreier graphs explicitly in thelater sections.PROPOSITION 1. Let  G be a finitely generated group,  S be a symmetric finitegenerating set of  G and  M be a set. Let  \varphi :  Garrow Aut(M) be a homomorphism.For an element  x_{0}\in M the Schreier graph (Orb  (x_{0}),  S,  \varphi,  x_{0} ) with the marked vertex x_{0} is  S ‐isomorphic to the Schreier coset graph  (G/H, S, H) with the marked vertex H=Stab(x_{0}) as marked labelled directed graphs.233. The Schreier graph of the action  \phi_{x}We consider the Baumslag‐Soliter group   BS(1, n)=\{\{a, b\}\rangle , where  n\geq 2.For any  x  \in \mathbb{R} the inclusion  \rho :  \{\{a, b\}\rangle  \mapsto Aut (  \mathbb{R} ) induces the action  \phi_{x} : \{\{a, b\}\rangle  arrow Aut(Orb  (x) ) given by  \phi_{x}(g)  =\rho(g)|  =g| . We will consider theSchreier graph (Orb  (x),  \{a,  b\}^{\pm},  \phi_{x} ). From now on, this Schreier graph is denotedby (Orb  (x),  \{a,  b\}^{\pm} ).Let  X=\{0,1, n-1\} . The set of all finite words over  X and the set of allinfinite words over  X are denoted by  X^{*} and  X^{\omega} respectively. Let  \tilde{X}=X^{*}\backslash \{\varepsilon\},where  \varepsilon denotes the empty word. For a word  w=w_{1}w_{2}\ldots w_{n} in  X^{*} the length of theword  w , denoted by  1h(w) , is the number  n . Note that the length of the empty word \varepsilon is zero.Let  \sigma :  X^{\omega}arrow X^{\omega} be the sift map given by  \sigma(w_{1}w_{2}w_{3}\ldots)=w_{2}w_{3}w_{4} Thei‐th letter of the infinite word  \sigma^{k-1}(w) , where  k\geq 1 and  w\in X^{\omega} is denoted by \sigma^{k-1}(w)_{i}(=w_{k-1+i}) .For any  v\in X^{\omega} put  D_{v}= \mathbb{Z}+\sum_{i\geq 1}v_{i}/n^{i},  D_{v}^{t}=n \mathbb{Z}+t+\sum_{i\geq 1}v_{i}/n^{i} , where t\in X . Note that  0 \leq\sum_{i\geq 1}v_{i}/n^{i}\leq 1 and  D_{v}=\sqcup_{t\in X}D_{v}^{t}.Let  w\in X^{\omega} . Put M_{w}= \bigcup_{j\geq 1}D_{\sigma^{j}(w)}\cup\bigcup_{u\in x*}D_{uw}\cup\bigcup_{j\geq 1}\bigcup_{u\in x*}\bigcup_{t\in X,t\neq w_{j}}D_{ut\sigma^{j}(w)}.For any  x\in \mathbb{R} there exist  y\in[0,1 ) and  n\in \mathbb{Z} such that  x=b^{n}(y) and the orbitOrb  (x) equals the orbit Orb (y) . Thus it suffices to consider only the Schreier graph(Orb  (y),  \{a,  b\}^{\pm} ) for  y\in[0,1 ).PROPOSITION 2. Suppose that  y\in[0,1 ) can be written by  y= \sum_{i\geq 1}w_{i}/n^{i} forsome  w\in X^{\omega} . Then the orbit Orb (y) coincides with the set  M_{w}.4. The Schreier graph of the action  \phi_{q}We say that a pair  (x, y) of words satisfies (A) if  x\in X^{*} and  y\in\tilde{X} satisfy thefollowing two conditions.(1) For any  k\geq 2 and any  t\in\tilde{X},  y\neq t^{k}.(2)  x\neq\varepsilon\Rightarrow x_{1h(x)}\neq y_{1h(y)}.In this section we will construct Schreier graphs for all rational numbers.Let  q be a rational number in  \mathbb{Q}\cap[0,1 ). Then there exist words  u\in X^{*} and v\in\tilde{X} such that  q= \sum_{i\geq 1}(uv^{\infty})_{i}/n^{i},  1h(v)\geq 1 , and the pair  (u, v) satisfies (A) .THEOREM 1. The Schreier graph (Orb  (q),  \{a,  b\}^{\pm} )  =(M_{uv}\infty, \{a, b\}^{\pm}) has thefollowing structure.24TAKAMICHI SATO(1) The set   M_{uv}\infty coincides with the following set  \tilde{M}_{uv}\infty , where \tilde{M}uv  \infty=lh  (u)+1h(v)\square -1_{D_{\sigma^{j}(uv^{\infty})}}\sqcup lh  (u)+1h\square (  v )  \lfloor\rfloor  \lfloor\rfloor  D_{st\sigma^{j}(uv^{\infty})}. j=1h(u) j=1h(u)+1s\in X^{*}t\in X, t\neq(uv^{\infty})_{j}(2) For any rational element  w in  X^{\omega} with any one of the forms  \sigma^{j}(uv^{\infty}) and st\sigma^{j}(uv^{\infty}) , there is a labelled directed graph consisting of  D_{w} and  D_{w}\cross\{b\}^{\pm}as the set of vertices and edges respectively. For any edge  e in  D_{w}\cross\{b\}^{\pm} withthe label  b , there exists an integer  j\in \mathbb{Z} such that   \alpha(e)=j+\sum_{i\geq 1}w_{i}/n^{i} and  \beta(e)=j+1+\sum_{i\geq 1}w_{i}/n^{i}.(3) For any rational element  w in  X^{\omega} with any one of the forms  \sigma^{j}(uv^{\infty}) and st\sigma^{j}(uv^{\infty}) , there exists the set of edges  D_{w}\cross\{a\} labelled by  a such that eachelement in the set  D_{w} is the initial vertex of an edge in  D_{w}\cross\{a\} and the terminalvertex of the edge lies in  D_{\sigma(w)}^{w_{1}}\subset D_{\sigma(w)}.  D_{\sigma(w)}^{w_{1}}\cross\{a^{-1}\} is the set of inverseedges of  D_{w}\cross\{a\}.(4) For any rational element  w in  X^{\omega} with any one of the forms  \sigma^{j}(uv^{\infty}) and st\sigma^{j}(uv^{\infty}) , D_{w}=\lfloor\rfloor D_{w}^{t}t\in X^{\cdot}5. The Schreier graph of the action  \phi_{\alpha}An element  w\in X^{\omega} is called a rational element in  X^{\omega} if there exist  u\in X^{*} and v\in\tilde{X} such that  w=uv^{\infty} . An element  w\in X^{\omega} is called an irrational element in  X^{\omega}if  w is not a rational element in  X^{\omega} . Let  \alpha be an irrational number in  (\mathbb{R}\backslash \mathbb{Q})\cap[0,1 ).Then there exists an irrational element  w\in X^{\omega} such that   \alpha=\sum_{i\geq 1}w_{i}/n^{i}.In this section we will describe the Schreier graph (Orb  (\alpha),  \{a,  b\}^{\pm} )  = (M_{w}, \{a, b\}^{\pm}) . We notice that the Schreier graph is  \{a, b\}^{\pm} ‐isomorphic to the Cayleygraph of   BS(1, n)=\{\{a, b\}\rangle relative to the generators  \{a, b\}^{\pm} by Proposition 1 sincethe stabilizer of  \alpha is trivial. However in the previous section we have constructedthe Schreier graphs (Orb  (q),  \{a,  b\}^{\pm} ) for rational elements  q and will compare thosedescriptions in the later section (see Theorem 2). Therefore we employ the Schreiergraph (Orb  (\alpha),  \{a,  b\}^{\pm} ). We construct the Schreier graph (Orb  (\alpha),  \{a,  b\}^{\pm} ) by an ar‐rangement of elements in the orbit Orb  (\alpha) . The construction of the Cayley graphof   BS(1, n)=\{\{a, b\}\rangle given in [4] depends on the fact that the word problem for BS(1, n) is solvable.PROPOSITION 3. The Schreier graph (Orb  (\alpha),  \{a,  b\}^{\pm} )  =(M_{w}, \{a, b\}^{\pm}) has thefollowing structure.25(1) The set  M_{w} coincides with the disjoint union j\geq 1u\in X^{*}j\geq 1u\in X^{*}t\in X,t\neq w_{j}\square  D_{\sigma^{j}(w)}\sqcup\lfloor\rfloor D_{uw}\sqcup\square \lfloor\rfloor\lfloor\rfloor D_{ut\sigma^{j}(w)}.(2) For any irrational element  v in  X^{\omega} with any one of the forms  \sigma^{j}(w) , uw, and ut\sigma^{j}(w) , there is a labelled directed graph consisting of  D_{v} and  D_{v}\cross\{b\}^{\pm} asthe set of vertices and edges respectively. For any edge  e in  D_{v}\cross\{b\}^{\pm} withthe label  b , there exists an integer  j\in \mathbb{Z} such that   \alpha(e)=j+\sum_{i\geq 1}v_{i}/n^{i} and  \beta(e)=j+1+\sum_{i\geq 1}v_{i}/n^{i}.(3) For any irrational element  v in  X^{\omega} with any one of the forms  \sigma^{j}(w) , uw, and ut\sigma^{j}(w) , there exists the set of edges  D_{v}\cross\{a\} labelled by  a such that each elementof the set  D_{v} is the initial vertex of an edge in  D_{v}\cross\{a\} and the terminal vertexof the edge lies in the set  D_{\sigma(v)}^{v_{1}}\subset D_{\sigma(v)}.  D_{\sigma(v)}^{v_{1}}\cross\{a^{-1}\} is the set of inverseedges of  D_{v}\cross\{a\}.(4) For any irrational element  v in  X^{\omega} with any one of the forms  \sigma^{j}(w) , uw, and ut\sigma^{j}(w) , D_{v}=\lfloor\rfloor D_{v}^{t}t\in X^{\cdot}6. ApplicationsIn this section, first we classify Schreier graphs described in the previous sections.THEOREM 2. Let  S=\{a, b\}^{\pm}.(1) For any irrational numbers  \alpha_{1},  \alpha_{2}\in \mathbb{R}\backslash \mathbb{Q} the Schreier graph (Orb  (\alpha_{1}),  S,  \alpha_{1} ) is S ‐isomorphic to the Schreier graph (Orb  (\alpha_{2}),  S,  \alpha_{2} ) as marked labelled directedgraphs.(2) For any rational number  q\in \mathbb{Q} and any irrational  number\alpha\in \mathbb{R}\backslash \mathbb{Q} the Schreiergraph (Orb  (q),  S ) is not isomorphic to the Schreier graph (Orb  (\alpha),  S ) as labelleddirected graphs.(3) Let  q_{1},  q_{2} be any rational numbers in  \mathbb{Q} . Suppose that there exist  m_{i}\in \mathbb{Z}, u_{i}\in X^{*} , and  v_{i}\in\tilde{X} such that  q_{i}=m_{i}+ \sum_{j\geq 1}(u_{i}v_{i}^{\infty})_{j}/n^{j} for each  i , where thepair  (u_{i}, v_{i}) satisfies (A) . Then the following conditions are equivalent.(a) The Schreier graph (Orb  (q_{1}),  S ) is isomorphic to the Schreier graph(Orb  (q_{2}),  S ) as labelled directed graphs.(b) Orb  (q_{1})=Orb(q_{2}) or Orb  (-q_{1})=Orb(q_{2}) .(c) There exists a nonnegative integer  j with  j<1h(v_{1}) such that  v_{2}^{\infty}= \sigma^{j}(v_{1}^{\infty}) or there exists a nonnegative integer  j with  j<1h(v_{1}) such that26TAKAMICHI SATO v_{2}^{\infty}=\sigma^{j}(\overline{v_{1}}^{\infty}) , where put  \overline{t}=n-1-t for  t\in X and  \overline{v}=\overline{v_{1}}\ldots\overline{v_{1h(v)}} for v\in\tilde{X}.COROLLARY 1. Let  S=\{a, b\}^{\pm} . Let  q_{1},  q_{2} be any rational numbers in  \mathbb{Q} . Thenthe followings are equivalent.(a) The Schreier graph (Orb  (q_{1}),  S,  q_{1} ) is isomorphic to the Schreier graph(Orb  (q_{2}),  S,  q_{2} ) as marked labelled directed graphs.(b)  |q_{1}|=|q_{2}|.By noting a closed edge path in the Schreier graph (Orb  (q),  S,  q ) which has anon‐trivial sequence of labels in  BS(1, n) , we have next proposition.PROPOSITION 4. For any rational number  q\in \mathbb{Q} the stabilizer Stab (q) is iso‐morphic to  \mathbb{Z}.Next we introduce the definition of presentation isomorphic subgroups in order totranslate the graphical expression of the Schreier graphs into the algebraic expressionof subgroups. Consequently, we get a relevance to presentations for the stabilizers fromthe previous result about the classification of the Schreier graphs(see Proposition 6).For any  i\in\{1,2\} let  G_{i} be a group and  T_{i} be a generating set of  G_{i} . Let T_{i}^{-1}=\{t^{-1}|t\in T_{i}\} and  T_{i}^{\pm}=T_{i}\cup T_{i}^{-1} We assume that (*) t\in T_{i}\cap T_{i}^{-1}\Leftrightarrow t\in T_{i}, t^{2}=1.For any  i\in\{1,2\} let  X_{i}=\{x_{t}|t\in T_{i}\} . Put  X_{i}^{-1}=\{x_{t}^{-1}|t\in T_{i}\} , where  x_{t}^{-1}denotes a new symbol corresponding to the element  x_{t} . We assume that   X_{i}\cap X_{i}^{-1}=\emptysetand that the expression  (x_{t}^{-1})^{-1} denotes the element  x_{t} . For any  i\in\{1,2\} the freegroup with the basis  X_{i} is denoted by  F(X_{i}) , and for a subset  R_{i} of  F(X_{i}) the normalclosure of the set  R_{i} in  F(X_{i}) is denoted by  \{\langle R_{i}\rangle\} . Let  G_{i} have the presentation \{X_{i}|R_{i}\rangle with respect to the epimorphism  \psi_{i} :  F(X_{i})arrow G_{i} given by  \psi_{i}(x_{t})=t.DEFINITION 2. For any  i\in\{1,2\} let  H_{i} be a subgroup of  G_{i}.  H_{1} is presentationisomorphic to  H_{2} if there exists a bijection  \gamma :  X_{1}^{\pm}arrow X_{2}^{\pm} with  \gamma(x_{t}^{-1})=\gamma(x_{t})^{-1} suchthat  \tilde{\gamma}(\psi_{1}^{-1}(H_{1}))=\psi_{2}^{-1}(H_{2}) and  \tilde{\gamma}(\{\langle R_{1}\rangle\})=\{\langle R_{2}\rangle\} , where  \tilde{\gamma} :  F(X_{1})arrow F(X_{2})given by  \tilde{\gamma}  (x_{t_{1}}^{\varepsilon_{1}} . . . x_{t_{k}}^{\varepsilon_{k}})=\gamma(x_{t_{1}})^{\varepsilon_{1}}\cdots\gamma(x_{t_{k}})^{\varepsilon_{k}},  \varepsilon_{i}=\pm 1.PROPOSITION 5. Let  \Gamma_{i}=(G_{i}/H_{i}, T_{i}^{\pm}, H_{i}),  t_{j}\in T_{1} , and  \varepsilon_{j}=\pm 1 . Then thefollowings are equivalent.(a)  \Gamma_{1} is isomorphic to  \Gamma_{2} as marked labelled directed graphs by a bijection  \gamma :   T_{1}^{\pm}arrow T_{2}^{\pm} satisfying the condition(  C )  t_{1}^{\varepsilon_{1}}\cdots t_{k}^{\varepsilon_{k}}=1_{G_{1}}\Leftrightarrow\gamma(t_{1}^{\varepsilon_{1}})\cdots\gamma(t_{k}^{\varepsilon_{k}})=1_{G_{2}}.(b)  H_{1} is presentation isomorphic to  H_{2}.27By Proposition 5 and Corollary 1, we obtain the following proposition.PROPOSITION 6. Let  q_{1},  q_{2}\in \mathbb{Q} . Then the followings are equivalent.(a) Stab  (q_{1}) is presentation isomorphic to Stab  (q_{2}) .(b)  |q_{1}|=|q_{2}|.References[1] G. BAUMSLAG, D. SOLITAR, Some two‐generator one‐relator non‐Hopfian groups, Bull. Amer.Math. Soc. 68 (1962), 199‐201.[2] B. FARB, L. MOSHER, A rigidity theorem for the solvable Baumslag‐Solitar groups, with anappendix by Daryl Cooper, Invent. Math. 131 (1998), no. 2, 419‐451.[3] B. FARB, L. MOSHER, Quasi‐isometric rigidity for the solvable Baumslag‐Solitar groups II,Invent. Math. 137 (1999), no. 3, 613‐649.[4] J. MEIER, Groups, graphs and trees, An introduction to the geometry of infinite groups, LondonMathematical Society Student Texts 73, Cambridge University Press, Cambridge, 2008.[5] D. SAVCHUK, Schreier graphs of actions of Thompson’s group F on the unit interval and on theCantor set, Geom. Dedicata 175 (2015), 355‐372.[6] D. SAVCHUK, Some graphs related to Thompson’s group F, Combinatorial and geometric grouptheory, Trends Math. Birkhäuser/Springer Basel AG, Basel, 2010, 279‐296.Present Address:TAKAMICHI SATOGRADUATE SCHOOL OF FUNDAMENTAL SCIENCE AND ENGINEERING,WASEDA UNIVERSITY,3‐4‐1 OKUBO, SHINJUKU‐KU, TOKYO, 169‐8555, JAPAN.e‐mail: improvement‐tak@ruri.waseda.jp

Schreier coset graphs of Baumslag-Solitar groups (Advances in General Topology and their Problems)

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Schreier coset graphs of Baumslag-Solitar groups (Advances in General Topology and their Problems)

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