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Schreier coset graphs of Baumslag‐Solitar groups
Takamichi Sato

Graduate School of Fundamental Science and Engineering, Waseda University

1.

Introduction

This article is an extended version of the talk given at the RIMS Meeting on Set
Theoretic 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 thatBS(3,2) is the first example of non‐Hopfian groups with one defining relation. Since
then these groups have served as a proving ground for many new ideas in combinatorial

and geometric group theory (see for example, [2, 3]).
Schreier coset graphs are generalizations of the Cayley graph of a group G , which
are 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. However
once 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 constructed
Schreier graphs of Thompson’s groupthe group.

F

from a motivation to study amenability ofIn this article, we focus on the group BS(1, n) , where n\geq 2 , which has an action
on the real line, and we explicitly construct Schreier graphs of the group for stabilizers
of all points under this action. As its consequence, we classify the Schreier graphs up
to isomorphism. This leads to a relevance to presentations for the stabilizers which

turn out to be infinite index subgroups in the group BS(1, n) .
2.

Definitions and notations

Let

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 byBS(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} by22
TAKAMICHI 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 setof 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 to
be 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 isomorphic
to \Gamma_{2} if \Gamma_{1} is isomorphic to \Gamma_{2} as labelled directed graphs and the mapping between
vertices preserves the marked vertices.
Let S be a generating set of a group G . The generating set S is symmetric if
V

S=S^{-1}.

DEFINITION 1. Let G be a finitely generated group, S be a symmetric finite
generating 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 andG/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 finite
generating 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 vertexx_{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.

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3.

The Schreier graph of the action \phi_{x}BS(1, n)=\{\{a, b\}\rangle , where n\geq 2.
For any
the inclusion \rho : \{\{a, b\}\rangle
Aut ( \mathbb{R} ) induces the action \phi_{x} :arrow\{\{a, b\}\rangle
Aut(Orb (x) ) given by \phi_{x}(g) =\rho(g)| =g| . We will consider the
Schreier 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 all
We consider the Baumslag‐Soliter groupx\in \mathbb{R}\mapstoinfinite 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}i‐th letter of the infinite word

\sigma^{k-1}(w)_{i}(=w_{k-1+i})\sigma^{k-1}(w) , where k\geq 1 andw\in X^{\omega}The

is denoted by.

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
0 \leq\sum_{i\geq 1}v_{i}/n^{i}\leq 1 and D_{v}=\sqcup_{t\in X}D_{v}^{t}.

For any v\in X^{\omega} put
t\in X . Note that

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 orbit
Orb (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},(2) x\neq\varepsilon\Rightarrow x_{1h(x)}\neq y_{1h(y)}.

y\neq t^{k}.

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^{*} andv\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.
following structure.

The Schreier graph (Orb (q), \{a, b\}^{\pm} ) =(M_{uv}\infty, \{a, b\}^{\pm}) has the24
TAKAMICHI SATO

(1) The set M_{uv}\infty coincides with the following set \tilde{M}_{uv}\infty , wherelh\tilde{M}uv\infty=lh(u)+1h(v)\square -1_{D_{\sigma^{j}(uv^{\infty})} \sqcup (u)+1h\square( v)\lf o r\rflo r\lf o r\rflo r

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}) andst\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}) andst\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 terminal
vertex 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 inverse

edges of D_{w}\cross\{a\}.
(4) For any rational element w in X^{\omega} with any one of the forms \sigma^{j}(uv^{\infty}) andst\sigma^{j}(uv^{\infty}),

D_{w}=\lf o r\rflo r D_{w}^{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^{*} andv\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 Cayley
graph of BS(1, n)=\{\{a, b\}\rangle relative to the generators \{a, b\}^{\pm} by Proposition 1 since
the stabilizer of \alpha is trivial. However in the previous section we have constructed
the 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 forBS(1, n) is solvable.
PROPOSITION 3.
following structure.

The Schreier graph (Orb (\alpha), \{a, b\}^{\pm} ) =(M_{w}, \{a, b\}^{\pm}) has the

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\lf o r\rflo r D_{uw}\sqcup\square \lf o r\rflo r\lf o r\rflo r D_{ut\sigma^{j}(w)}.
(2) For any irrational element v in X^{\omega} with any one of the forms \sigma^{j}(w) , uw, andut\sigma^{j}(w) , there is a labelled directed graph consisting of D_{v} and D_{v}\cross\{b\}^{\pm} as
the 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, andut\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 inverse

edges of D_{v}\cross\{a\}.
(4) For any irrational element v in X^{\omega} with any one of the forms \sigma^{j}(w) , uw, andut\sigma^{j}(w),D_{v}=\lf o r\rflo r D_{v}^{t} \in X^{\cdot}
6.

Applications

In 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 directed
graphs.

(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 labelled
directed graphs.

(3) Let q_{1}, q_{2} be any rational numbers in \mathb {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 that

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TAKAMICHI 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)} forv\in\tilde{X}.
COROLLARY 1. Let S=\{a, b\}^{\pm} . Let q_{1}, q_{2} be any rational numbers in \mathb {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.

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}=\emptyset
and that the expression (x_{t}^{-1})^{-1} denotes the element x_{t} . For any i\in\{1,2\} the free
For any i\in\{1,2\} let X_{i}=\{x_{t}|t\in T_{i}\} . Putgroup with the basis X_{i} is denoted by F(X_{i}) , and for a subset R_{i} of F(X_{i}) the normal
closure 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 presentation

isomorphic 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} such\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{\gam a} : F(X_{1})arrow F(X_{2})given by \tilde{\gam a} (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.

that

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}ar ow

T_{2}^{\pm} satisfying the condition( C)

t_{1}^{\varepsilon_{1} \cdots t_{k}^{\varepsilon_{k} =1_{G_{1} \Leftrightar ow\gamma(t_{1}^{\varepsilon_{1} )\cdots\gamma(t_{k} ^{\varepsilon_{k} )=1_{G_{2} .

(b) H_{1} is presentation isomorphic to H_{2}.

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By Proposition 5 and Corollary 1, we obtain the following proposition.
PROPOSITION 6.

(a)
(b)

Let q_{1}, q_{2}\in \mathbb{Q} . Then the followings are equivalent.

Stab (q_{1}) is presentation isomorphic to Stab (q_{2}) .
|q_{1}|=|q_{2}|.
References

[1]
[2]
[3]
[4]

G. BAUMSLAG, D. SOLITAR, Some two‐generator one‐relator non‐Hopfian groups, Bull. Amer.
Math. Soc. 68 (1962), 199‐201.
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.
B. FARB, L. MOSHER, Quasi‐isometric rigidity for the solvable Baumslag‐Solitar groups II,
Invent. Math. 137 (1999), no. 3, 613‐649.
J. MEIER, Groups, graphs and trees, An introduction to the geometry of infinite groups, London
Mathematical Society Student Texts 73, Cambridge University Press, Cambridge, 2008.

[5]
[6]

D. SAVCHUK, Schreier graphs of actions of Thompson’s group F on the unit interval and on the
Cantor set, Geom. Dedicata 175 (2015), 355‐372.
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