数理解析研究所講究録
第 1911 巻 2014 年 57-75

57

Concerning actions of 3-manifold groups:
from topological and arithmetic viewpoints
Takashi HARA
Department of $Mathematics_{\}}$
School of Science and Technology for Future Life,
Tokyo Denki University
Abstract

After reviewing Marc Culler and Peter B. Shalen’s construction of essential surfaces contained in
3-dimensional manifolds, we shall present an extension of there construction utilising character varieties and Bruhat-Tits buildings of higher dimension. This extended method enables us to constructessential tribranched surfaces contained in 3-dimensional manifolds in a systematic way.

This is a report of the author’s talk “Concerning actions of 3-manifold groups: fromtopological and arithmetic viewpoints”’ given at the RIMS Conference: intelligence of
Low-dimensional Topology 2014. In 1983, Marc Culler and Peter B. Shalen proposed asystematic method to construct essential surfaces contained in 3-dimensional manifolds in[CS83]. Their method heavily utilised algebro-geometric ideas; to sum up, they introduced
geometry of $SL(2)$ -character varieties and the theory of Bruhat-Tits trees to establish theirconstruction. It should therefore be worth introducing more sophisticated algebraic andarithmetic concepts to extending Culler and Shalen’s classical method. In this brief report
we shall propose a systematic construction of essential tribranched surfaces contained in
3-manifolds, which is regarded as a natural extension of Culler and Shalen’s result [CS83](this is ajoint work with Takahiro KITAYAMA, Tokyo Institute of Technology). Refer also
to [H14], which deals with the same contents as this report but written in Japanese.

The author is very grateful to the organisers of the RIMS Conference:Intelligence of Low-dimensional Topology, Professor Tomotada Ohtsuki and Takahiro Kitayama, for kindly inviting him to this great conference in topology and giving him theopportunity to talk there. In particular he would like to express his sincere gratitude to
Takahiro Kitayama, who is also the coauthor of [HK14], for many fruitful discussion andcomments on the contents of his talk.
Acknowledgement.

58

Introduction: Essential surfaces and group actions\S 1

Throughout this article all 3-dimensional manifolds are assumed to be compact, connected, irreducible and orientable. As an introduction of this report, we here review the(somewhat mysterious) relation between the topological concepts of essential surfaces andalgebraic actions of 3-manifold groups on trees.
First let us recall the definition of essential surfaces (see [Sh02, Definition 1.5.1] forexample).

Definition 1.1 (Essential surfaces). A $surface^{*1}S$ contained in a 3-manifold $M$ is said
to be essential if the following four conditions are fulfilled:(ES1) (incompressibility) for each connected component of , the canonical homomorphism $\pi_{1}(S_{i})arrow\pi_{1}(M)$ is injective;(ES2) (bicollaredness) is bicollared; that is, there exists a homeomorphism from
$S\cross[-1, 1]$ onto a neighbourhood ofin $M$ such that$-h(x, 0)=x$ holds for each in ; andcoincides with $h(\partial S\cross[-1,1$
the intersection of $h(S\cross[-1,1])$ and(ES3) (non-boundary-parallel) no connected component of is boundary-parallel;(ES4) (nontriviality) is nonempty and no connected component of is homeomorphic to a 2-sphere.
$S$$S_{i}$

$h$

$S$

$S$

$x$

$S$

$\partial M$

-

$S$

$S$

$S$

Now let be an essential surface
contained in a 3-manifold $M$ andletdenote the (finite) set of
all connected components of . By
cutting $M$ along , we may decompose $M$ into several components,,...,
which we denote by
Now let us shrink each component
to a point (or a (vertex) and
Figure 1. The dual graph associated to an essential surface
label it . Whenever a connectedconan “edge”, we attach toandcomponentof separates components
and . As a result we obtain a finite graph, which is called thenecting the vertices(see Figure 1 above). We finallydual graph of $M$ associated to the essential surfaceof the correof the dual graph the fundamental groupattach to each vertex$S$$\{S_{j}\}_{j=1}^{r}$

$S$

$S$

$N_{s}.$$N_{1}$$N_{i}$$v_{i}$$S_{j}$$S$$N_{i_{1}}$$N_{i_{2}}$$S_{i}$$v_{i_{2}}$$v_{i_{1}}$$S$$\pi_{1}(N_{i})$$v_{j}$$*1$

Here we do not require that$S$

itself$1S$

connected.

$e_{j}$

59$\exists_{\tilde{f}}$$\tilde{M}$$\mathcal{T}$$\pi_{1}(M)-equivariant|$$|$

quotient$\mathcal{T}/\pi_{1}(M)$Figure 2. Construction of (essential) surfaces

sponding component(which is called the vertex group of), and attach to each edge
the fundamental groupof the corresponding component(which is called theedge group of ). After these procedures we obtain a graph of groups
associated to $S$$N_{i}$$v_{i}$$\pi_{1}(S_{j})$$e_{j}$$S_{j}$$e_{j}$$\mathcal{G}s$(see [Se77, Chapitre I,, Definition 8] for the definition of graphs of groups). Due to thetheory of graphs of groups developed by Hyman Bass and Jean-Pierre Serre (for details
refer to [Se77, Chapitre I
the graph of groups
is known to be obtained as the quotient graph associated to an action of $\pi_{1}(M)$ on a certain connected and simply connected
graph, or namely, on a certain. Each vertex and edge group is recovered (up to
conjugation) as the isotropy subgroup of $\pi_{1}(M)$ (with respect to the action of $\pi_{1}(M)$ on
at the corresponding vertex and edge respectively.
$\mathcal{G}s$$tree^{*2\prime}\mathcal{T}$$*3$$\mathcal{T})$Conversely, given an action of
on a tree which is nontriviar4 and without inversion, we may construct an essential surface contained in $M$ . This is due to results of$\pi_{1}(M)$

$*5$

$S$

$*2$

$*3$

This tree is an analogous object of the universal covemng space in the theory of covering spaces, and called theBass-Serre tree today. See [Se77, Chapitre I., Section 53] for details.
Moreover we may obtain the splitting of $\pi_{1}(M)$ as the fundamental groupof the graph of groups
An action of a groupon a tree
is said to be nontrivial if the isotropy subgroupdoes not coincide with thewhole groupfor each vertex
An action of a groupon a tree is said to be without inversion if there does not exist an element of which
fix an edge but interchange the endpoints of$\pi_{1}(\mathcal{G}_{S})$

$*4$

$G$

$G$

$*5$

$\mathcal{G}s.$$G_{v}$

$v.$

$G$

$e$

$\mathcal{T}$$\mathcal{T}$

$g$

$e.$

$G$

60
many people including John Robert Stallings [St59, St71], David Bernard Alper Epstein[Ep61] and Riedhelm Waldhausen [Wa67]. We here briefly sketch how to construct anessential surface from a group action on a tree (refer also to [Sh02, Section 2.2.]). First
take a triangulation of $M$ and lift it to the universal covering spaceof $M$ . Then we
obtain a $\pi_{1}(M)$ -equivariant triangulation of . Due to the contractibility of the tree
we may construct a-equivariant simplicial map(by using simplicial
approximation theorem if necessary). The simplicial map induces a piecewise-linear
map $f:Marrow \mathcal{T}/\pi_{1}(M)$ on the quotient, where
denotes the quotient graph ofwith respect to the action of $\pi_{1}(M)$ on . Let denote the set of the midpoints of all, and consider the inverse image $S’$ of under theedges of the quotient graph
map $f:Marrow \mathcal{T}/\pi_{1}(M)$ . The inverse image $S’$ is then a properly embedded surface in $M$(see Figure 2). While $S’$ itself might not be essential, we may modify
into an essential
surface by certain “local surgery”’ (that is, a certain topological operation). We may
thus construct an essential surface in $M$ from the nontrivial action of $\pi_{1}(M)$ on thetree$\tilde{M}$$\tilde{M}$$\mathcal{T},$$\tilde{f}:\tilde{M}arrow \mathcal{T}$$\pi_{1}(M)$$\tilde{f}$$\mathcal{T}/\pi_{1}(M)$$\mathcal{T}$$E$$\mathcal{T}$$E$$\mathcal{T}/\pi_{1}(M)$

$S’$

$S$

$S$

$\mathcal{T}.$As a consequence, one observes that there exists a close relation between the conceptsof essential surfaces and (nontrivial) actions of 3-dimensional groups on trees:
Essential surfaces(topological concept)$clos\underline{erela}$tion

Nontrivial actions of $\pi_{1}(M)$ on trees(algebraic group action)

Moreover it is easy to imagine from the arguments above that one might be able tosystematically construct essential surfaces contained in 3-manifolds if one could constructnontrivial actions of 3-manifold groups on trees in a certain ingenious manner. It was
Marc Culler and Peter B. Shalen who realised this vague idea and established a systematicconstruction of essential surfaces contained in 3-manifolds. In the next section we shallsurvey their outstanding results of [CS83].

\S 2

Nontrivial actions of 3-manifold groups: Culler-Shalen theory

In their construction of nontrivial actions of$\pi_{1}(M)$on trees, Culler and Shalen utilised

two highly algebraic (or algebro-geometric) ingredients: namely--

Geometry of $SL(2)$ -character varieties; andBruhat-Tits trees associated to the special linear groups of degree two.

We shall see each of them more thoroughly in the following subsections.

61\S 2.1

Step 1: Geometry of $SL(2)$ -character varieties

Concerning $SL(2)$ -character varieties, see also Harada’s article [Harada14] in this volume. Let us set$R(\pi_{1}(M))_{SL(2)}(\mathbb{C})=Hom(\pi_{1}(M), SL_{2}(\mathbb{C}))$$=$(the set of all$SL_{2}(\mathbb{C})$-representations of $\pi_{1}(M)$ ).

Using the fact thatis a group of finite presentation (due to the compactness of, we may readily verify that
admits a structure of an aﬃne algebraic variety over , which we call the $SL(2)$ -representation variety of $\pi_{1}(M)$ . Next let
denote the set of all $SL(2)$ -characters of; more specifically, anelement ofis the traceof a certain $SL(2)$ -representationof. It is widely known that
also admits astructure of an aﬃne algebraic variety over , which we call the $SL(2)$ -character varietyof.
In the context of algebraic geometry, the natural action of
on the set
via conjugation
canonically extends to the action of the(reductive) algebraic groupon the representation variety
and the character variety
is represented as the (geometric invariant$\pi_{1}(M)$$M)$$R(\pi_{1}(M))_{SL(2)}(\mathbb{C})$$\mathbb{C}$$X(\pi_{1}(M))_{SL(2)}(\mathbb{C})$$\pi_{1}(M)$$X(\pi_{1}(M))_{SL(2)}(\mathbb{C})$$\rho:\pi_{1}(M)arrow SL_{2}(\mathbb{C})$$\chi_{\rho}=Tr(\rho)$$\pi_{1}(M)$$X(\pi_{1}(M))_{SL(2)}(\mathbb{C})$$\mathbb{C}$$\pi_{1}(M)$$SL_{2}(\mathbb{C})$$R(\pi_{1}(M))_{SL(2)}(\mathbb{C})$$\rho\mapsto\gamma^{-1}\rho\gamma$$SL(2)/\mathbb{C}$$R(\pi_{1}(M))_{SL(2)}/\mathbb{C}$’$X(\pi_{1}(M))_{SL(2)}/\mathbb{C}$theoretical) quotient variety ofwith respect to this action. In particular$R(\pi_{1}(M))_{SL(2)}/\mathbb{C}$there exists a natural “quotient” morphism$\pi:R(\pi_{1}(M))_{SL(2)}/\mathbb{C}arrow X(\pi_{1}(M))_{SL(2)}/c.$Let$\gamma$be an element of $\pi_{1}(M)$ . Then we associate tosatisfying$\gamma$a regular function$I_{\gamma}$

defined on$X(\pi_{1}(M))_{SL(2)}/\mathbb{C}$$I_{\gamma}(\chi_{\rho})=Tr\rho(\gamma)$

for every complex pointof(the $evaluation-at-\gamma$ ” function). It is
known that the aﬃne coordinate ring ofis generated over by a finite
number of such evaluation functions (see [CS83, Proposition 1.4.1]).
Now let us take an (aﬃne) algebraic curve
contained in. By astandard argument in algebraic geometry (refer to [Mu76] for basic facts on algebraic geometry), one may verify that there exists a “lift” $D$ of with respect to ; in other words,
there exists an algebraic curve contained insuch that the restriction
$D$
of to
(which we also abbreviate as ) surjects onto . The canonical projection$\pi:Darrow C$ extends to their projective completions, and induces a regular morphism$\chi_{\rho}$$X(\pi_{1}(M))_{SL(2)}/\mathbb{C}$$\mathbb{C}$$X(\pi_{1}(M))_{SL(2)}/\mathbb{C}$$C$$X(\pi_{1}(M))_{SL(2)}/\mathbb{C}$

$C$

$D$

$\pi$

$\pi$

$R(\pi_{1}(M))_{SL(2)}/\mathbb{C}$

$\pi$

$C$

62
$\tilde{\pi}:\tilde{D}arrow\tilde{C}$$C$

on the (unique) smooth projective models of (the projective completions of)and $D.$For such $D$ we may consider the tautological representation$\rho_{taut,D}:\pi_{1}(M)arrow SL_{2}(\mathbb{C}[D])$wheredenotes the ring of regular functions on . This is a certain kind of (universal
representations and characterised by the following property: for every closed point ofand for every element of, we have$D$$\mathbb{C}[D]$

$y$

$D$

$\pi_{1}(M)$$\gamma$$\rho_{taut,D}(\gamma)(y)=\rho_{y}(\gamma)$

where denotes the-representation of $\pi_{1}(M)$ corresponding to the closed point
regarded as a complex point of. Composing it with the canonical maps*6$SL_{2}(\mathbb{C})$$\rho_{y}$$y$$R(\pi_{1}(M))_{SL(2)}/\mathbb{C}$$\mathbb{C}[D]\mapsto \mathbb{C}(D)arrow\sim \mathbb{C}(\tilde{D})$(here we denote by
andtively), we may readily extend$\mathbb{C}(D)$$\mathbb{C}(\tilde{D})$respecthe fields of rational functions on
andto the tautological representation associated to ;$D$$\tilde{D}$$\rho_{taut,D}$$\rho_{taut,\tilde{D}}:\pi_{1}(M)arrow SL_{2}(\mathbb{C}(\tilde{D}))$\S 2.2$\tilde{D}$.

Step 2: Bruhat-Tits trees associated to $SL(2)$The tree on which we let 3-manifold groups act is the Bruhat-Tits tree associated to thespecial linear group $SL(2)$ defined over a discrete valuation field (refer to [Se68, Chapitre I,
Section 1] for details on discrete valuation fields). We here recall its precise definition.
Let $K=(K, v)$ be a field equipped with (normalised) discrete valuationandits valuation ring. We denote by(that is, an element ofa uniformiser ofsatisfying $v(\varpi_{v})=1$ ). Let
be the standard 2-dimensional $K$ -vector space with basis. Recall that an -lattice of is a free -submodule ofwhich satisfies. Anofis said to be homothetic to another-lattice-lattice
if there exists an invertible element of $K$ such that(as ancoincides with
-submodule of ). One readily verifies that the homothety relation is an equivalence
relation on the set of all -lattices of$v:K^{\cross}arrow \mathbb{Z}$$a$$\mathcal{O}_{v}$$\mathcal{O}_{v}$$\mathcal{O}_{v}$$\varpi_{v}$$V_{0}$$\{e_{1}, e_{2}\}$$L$$\mathcal{O}_{v}$$L\otimes_{\mathcal{O}_{v}}K=V_{0}$$L_{1}$$\mathcal{O}_{v}$$L_{2}$$V_{0}$$V_{0}$$\mathcal{O}_{v}$$L_{1}$$a$$\mathcal{O}_{v}$$V_{0}$$\mathcal{O}_{v}$$V_{0}$$\mathcal{O}_{v}$$*6$$V_{0}.$The second isomorphism is due to the fact that $D$ and$\tilde{D}$are birational to each other.

$aL_{2}$

63
Definition 2.1 (The Bruhat-Tits tree). Let
be the set of all homothety classes$[L]$ of(the vertex set of-lattices of). We equipwith the following
adjacency relation:$V(\mathcal{T}_{K})$$\mathcal{O}_{v}$$V_{0}$$\mathcal{T}_{K}$$V(\mathcal{T}_{K})$two elements and ofare
if and only if there exist
andrepresenting
and respectively such that$v_{1}$$L_{1}$$L_{2}$$ad_{J^{acent}}$$V(\mathcal{T}_{K})$$v_{2}$$v_{1}$$\mathcal{O}_{v}$-lattices
holds.

$\varpi_{v}L_{1}\subset L_{2}\subset L_{1}$$v_{2}$

Let
denote the set of all adjacent pairsof(the edge set of).
Then one may verify that the resulted graphis indeed a tree(refer to [Se77, Chapitre II, Th\’eor\‘em 1 which we call the Bruhat-Tits tree associated
to $SL(2)_{/K}.$$E(\mathcal{T}_{K})$$(v_{1}, v_{2})$$V(\mathcal{T}_{K})$$\mathcal{T}_{K}$$\mathcal{T}_{K}=(V(\mathcal{T}_{K}), E(\mathcal{T}_{K}))$The figure on the right side illustrates the
Bruhat-Tits tree associated to the special linear group of degree two defined over the 2-adic
number field. Note that the Bruhat-Titstree
is equipped with the natural actionof $SL_{2}(K)$ ; the action of an elementof$SL_{2}(K)$ on the tree
is explicitly defined by$g*v=[9^{L}]$ when an-lattice of repreBruhat-Tits tree associated to $SL(2)$sents the vertex . It is known that this action
is strongly transitive and without inversion (refer to [Se77, Chapitre II, Section 1$\mathbb{Q}_{2}$$\mathcal{T}_{K}$$g$$\mathcal{T}_{K}$$L$$\mathcal{O}_{v}$$V_{0}$$-$$/\mathbb{Q}_{2}$$v$

We continue to use the notation used in the previous subsection. In order to relate$SL(2)$ -representations of the 3-manifold group $\pi_{1}(M)$ to the theory of Bruhat-Tits trees,
let us consider a closed point ofand take a lift of with respect to(more
specifically, is a closed point of satisfying). By elementary facts on algebraic
curves, each closed point of endow the rational function fieldof with a discrete
valuation*7 , and thus we may consider the Bruhat-Tits tree
associated to thespecial linear group $SL(2)$ defined over the discrete valuation field.
Composing the tautological representation$SL_{2}(K_{y})$with the canonical action of
on the Bruhat-Tits tree , we obtain the action of $\pi_{1}(M)$$\tilde{C}$$x$$\tilde{D}$$y$$\tilde{\pi}$

$x$

$y$

$\tilde{\pi}(y)=x$$\tilde{D}$$\tilde{D}$$\mathbb{C}(\tilde{D})$$y$$v_{y}$$\mathcal{T}_{y}=\mathcal{T}_{K_{y}}$$K_{y}=(\mathbb{C}(\tilde{D}), v_{y})$$\rho_{taut,\tilde{D}}$$\mathcal{T}_{y}$$\pi_{1}(M)arrow SL_{2}(K_{y})\rho_{taut,\overline{D}}arrow^{canonica1}$on the Bruhat-Tits tree associated to the closed point

$*7$

$y$

Aut

of

$(\mathcal{T}_{y})$$\tilde{D}.$The discrete valuation
is nothing but the ‘order at-y” function, namely, the evaluation ofdefined on
is the order of the Laurent expansion of around$v_{y}$$f$$\tilde{D}$

$f$

$y.$

$v_{y}$

at a rational function

64\S 2.3

Step 3: Construction of nontrivial actions

A closed point (resp. y) of (resp. ) is said to be an ideal point if it is containedin the $\langle(boundary$ ” of the projective completion of (resp. $D$ ). Or in other words, ideal
points are points added in the procedure of the projective completion. The key to theconstruction of nontrivial actions of $\pi_{1}(M)$ on the Bruhat-Tits trees is the following
theorem (denoted as the “Fundamental Theorem”’ in [CS83, Theorem 2.2.1]).
$\tilde{C}$$\tilde{D}$

$x$

$C$

We may prove Theorem 2.2 by a certain easy and tautological argument based uponis generated by evaluation
the fact that the aﬃne coordinate ring of. We omit the details of the proof, and the readers are referred tofunctions[CS83, Theorem 2.2.1] or [Sh02, Property 5.4.2 and Property 5.4.4] for details.
By virtue of Theorem 2.2, we obtain a nontrivial action (without inversion) of $\pi_{1}(M)$on a tree, and thus we may construct an essential surface contained in $M$ as explained in
the previous section. This is an outline of Culler and Shalen’s construction of essential
surfaces in [CS83].
$X(\pi_{1}(M))_{SL(2)}/\mathbb{C}$$\{I_{\gamma}\}_{\gamma\in\pi_{1}(M)}$\S 3

Extending towards representations of higher dimension

The content of this section is ajoint work with Takahiro KITAYAMA, Tokyo Institute ofTechnology. The details shall appear in [HK14].
As we have already seen in the previous section, Culler and Shalen’s construction of
essential surfaces is based upon the deep study of $SL(2)$ -representations of $\pi_{1}(M)$ . Therefore it is natural for us to consider the following naive question:
Naive Question:

Cannot we play the same game for representations ofdimension??

$\pi_{1}(M)$of higher

65
Or equivalently, are there any extension or any generalisation of Culler-Shalen theory
for representations ofof higher dimension? The main result of [HK14] presents anaﬃrmative answer to this naive question.
$\pi_{1}(M)$\S 3.1(Essential) Ribranched surfaces

The main diﬀerence between the main result of [HK14] from that of Culler and Shalenin [CS83] is that the former one provides not (usual) essential surfaces but essential
tribranched surfaces in general. In order to introduce the notion of essential tribranchedsurfaces, we first give the definition of tribranched surfaces.

In the definition above we denote by
the closed subset of which corresponds to thesubset) of) under the identification in (TBSO), by
the complementof a suﬃciently small tubular neighbourhoodofin , and by
the complement ofa suﬃciently small tubular neighbourhood ofin $M$ . We call the set
the branch setof . The figure on the right side shows how atribranched surface locally looks like.
$C(\Sigma)$$\Sigma$$\{0\}\cross[0, \infty$$\mathbb{C}\cross[0, \infty$$S(\Sigma)$$C(\Sigma)$$\Sigma$$\Sigma$$M(\Sigma)$$C(\Sigma)$$\Sigma$

A tribranched Surface$\Sigma$(local picture)

The essentiality of tribranched surfaces is defined similarly to the classical concept ofincompressibility (namely the condition (ES1) of Definition 1.1). The precise definitionof essential tribranched surfaces is given as follows.

66
Definition 3.2 (Essential tribranched surfaces). A tribranched surface
to be essential if it has following properties:$\Sigma$in $M$ is said

, the natural functorial homomorphism(ETBSI) for each component $N$ of$\pi_{1}(N)arrow\pi_{1}(M)$ is not surjective;,(ETBS2) for connected components $C,$
and $N$ of
andrespectively, the natural functorial homomorphisms $\pi_{1}(C)arrow\pi_{1}(S)$ and$\pi_{1}(S)arrow\pi_{1}(N)$ are injective (if they exist);(ETBS3) there does not exist a connected component of which is contained in aball in $M$ or a collar of$M(\Sigma)$$S$$C(\Sigma)$$S(\Sigma)$$M(\Sigma)$$\Sigma$$\partial M.$

The notion of (essential) tribranched surfaces is a certain extension of the notion ofusual (essential) surfaces; indeed usual (essential) surfaces may be regarded as (essential)
tribranched surfaces with the empty branch sets. (Essential) ‘nibranched surfaces seem,
however, to have mysterious and strange-looking properties which have not been observed
for usual (essential) surfaces. We shall propose several problems and questions concerning(essential) tribranched surfaces in [PLDT14, Section 10].
\S 3.2 Utilising Bruhat-Tits buildings of higher dimension

In order to pursue an extension of Culler and Shalen’s method, we should consider
counterparts of the following algebraic ingredients (used in [CS83]) for higher-dimensional
representations of $\pi_{1}(M)$ :the $SL(2)$ -character variety-the Bruhat-Tits tree-

$X(\pi_{1}(M))_{SL(2)}/\mathbb{C}$; and$\mathcal{T}_{\overline{y}}.$For the former one it is not diﬃcult at all to generalise the definitions of both thetorepresentation variety
and the character variety$SL(n)$ -representations of $\pi_{1}(M)$ ; indeed all one need to do is to replace(orandeverywhere in the definitions of
within Section 2.1. We may thus define the $SL(n)$ -representation variety
of $\pi_{1}(M)$ , and obtainand the $SL(n)$ -character variety
a canonical “quotient morphism$X(\pi_{1}(M))_{SL(2)}/\mathbb{C}$$R(\pi_{1}(M))_{SL(2)}/\mathbb{C}$$SL(2)_{/\mathbb{C}}$$SL_{2}(\mathbb{C}))$$SL(n)_{/\mathbb{C}}$$(or SL_{n}(\mathbb{C}))$$R(\pi_{1}(M))_{SL(2)}/\mathbb{C}$$X(\pi_{1}(M))_{SL(2)}/\mathbb{C}$$X(\pi_{1}(M))_{SL(n)}/\mathbb{C}$$R(\pi_{1}(M))_{SL(n)}/\mathbb{C}$$\pi:R(\pi_{1}(M))_{SL(n)}/\mathbb{C}arrow X(\pi_{1})_{SL(n)}/\mathbb{C}.$

Similarly to the $SL(2)$ -character variety, the aﬃne coordinate ring of the $SL(n)$ -characterby a finite number ofis known to be generated overvariety”$evaluation-at-\gamma$ functions, due to the classical result of Claudio Procesi [Pr76].
$\mathbb{C}$$X(\pi_{1}(M))_{SL(n)}/\mathbb{C}$$I_{\gamma}$

67
For the latter one –the Bruhat-Tits trees– there exist no trees which admit naturalactions of the special linear groupsof higher degree, but due to the vast theory
of Fran\caois Bruhat and Jacques Tits [BT72, BT84] on Euclidean buildings associated toreductive groups defined over local fields, there exists a higher-limensional (contractible)
simplicial complex equipped with the natural action of $SL(n)$ . It is called the Bruhat-Titsbuilding associated to $SL(n)$ .
The general definition of the Bruhat-Tits buildings is rather complicated, which we shallomit. Limiting ourselves to the special linear group $SL(n)$ , however, we may write down
the explicit construction of the Bruhat-Tits building associated to $SL(n)$ as follows (referalso to [Ga97, Chapter 19
As in Section 2.2, let $K=(K, v)$ be a discrete valuation
field andits valuation ring. Let us denote by
the standard -dimensional $K$-vector
space. We also consider-lattices ofand the homothety relation among them.
$S\dot{L}(n)$$\mathcal{O}_{v}$$n$$V_{0}$$V_{0}$$\mathcal{O}_{v}$Definition 3.3 (The Bruhat-Tits building associated $SL(n)$ ). Consider an abstract
simplicial complex
of dimension $n-1$ constructed in the following manner;$\mathcal{B}_{n,K}$-

the vertex set ofis defined as the set of all homothety classes of -lattices
in ; and(distinct) -vertices,...,
form a -simplex if and only if thereof, . . . , respectively such
exist -lattices,...,of representingthat$\mathcal{O}_{v}$$\mathcal{B}_{n,K}$$V_{0}$-$k$$v_{0},$$L_{0},$$\mathcal{O}_{v}$$L_{1}$$L_{k}$$k$$\mathcal{B}_{n,K}$$v_{k}$$v_{1}$$V_{0}$$v_{0},$$v_{k}$$v_{1}$$\varpi_{v}L_{0}\subset L_{k}\subset L_{k-1}\subset\cdots\subset L_{1}\subset L_{0}$holds (here$\varpi_{v}$denotes a uniformiser of $K$ ).

One may verify that the (abstract) simplicial complex
satisfies all the axioms of(thick) Euclidean buildings ( la Jacques Tits), which we call the Bruhat- Tits building
associated to $SL(n)_{/K}.$$\mathcal{B}_{n,K}$

We cite [AB08] and [Ga97] as basic references on (Euclidean) buildings. Combinatoriallythe (Euclidean) building
consists of the $(n-1)$ -dimensional (real) Euclidean spaces(called apartments of) tessellated by equilateral $(n-1)$ -dimensional simplices (called), which are glued together along (several) chambers. The right figurechambers ofof Table 1 illustrates an apartment and a chamber of the Bruhat-Tits buildings associatedto the special linear group of degree three. Due to the contractibility of every apartmentofand the axioms of buildings, we readily verify that the Bruhat-Tits building
is contractible. Note that the Bruhat-Tits building
associated to the special linear$\mathcal{B}_{n,K}$$\mathcal{B}_{n,K}$$*8$$\mathcal{B}_{n,K}$$\mathcal{B}_{n,K}$$\mathcal{B}_{n,K}$$\mathcal{B}_{2,K}$$*8$

The Bruhat-Tits building$\mathcal{B}_{n,K}$

is said to be Euclidean since each apartment of$\mathcal{B}_{n,K}$

is the Euclidean space.

68
group $SL(2)/K$ of degree 2 is none other than the Bruhat-Tits tree

associated toby definition. The left figure of Table 1 illustrates how the Bruhat-Tits trees are
regarded as Euclidean buildings.
$\mathcal{T}_{K}$$SL(2)_{/K}$

The Bruhat-Tits tree as an Euclidean building

An apartment and a chamber ofthe Bruhat-Tits building associated to $SL(3)$Table 1. The Bruhat-Tits buildings

As in the case for Bruhat-Tits trees, the Bruhat-Tits building
is equipped with thenatural action of $SL_{n}(K)$ ; namely an element of $SL_{n}(K)$ acts on a vertex ofby$g*v=[gL]$ when an-lattice of represents . This action is known to be strongly
transitive and type-preseruing.
$\mathcal{B}_{n,K}$

$v$

$g$

$L$

$\mathcal{O}_{v}$$V_{0}$$\mathcal{B}_{n,K}$

$v$

$*9$

Now let us take an (aﬃne) algebraic curve contained inand take$D$“lift”
ofwith respect to . The restriction of the natural quotient map to$D$extends to a regular morphismon the smooth projective models of (the$D$projective completions of) and , as in Section 2.1. Let be a closed point of andits lift to with respect to . Let
denote the discrete valuation of the field of rational
functionsof associated to the closed point and let us regard
as a discrete
valuation field. Then by composing the tautological representation
$C$

$C$

$a$

$X(\pi_{1}(M))_{SL(n)}/\mathbb{C}$

$\pi$

$\pi$

$\tilde{\pi}:\tilde{D}arrow\tilde{C}$$C$$\tilde{C}$

$x$

$y$

$\tilde{D}$$\tilde{\pi}$$v_{y}$$\mathbb{C}(\tilde{D})$$\tilde{D}$$\mathbb{C}(\tilde{D})$$y$$K_{y}=(\mathbb{C}(\tilde{D}), v_{y})$$\rho_{taut,\tilde{D}}:\pi_{1}(M)arrow SL_{n}(\mathbb{C}(\tilde{D}))=SL_{n}(K_{y})$with the natural action of $SL_{n}(K_{y})$ on the Bruhat-Tits building
action of $\pi_{1}(M)$$\mathcal{B}_{n,K_{y}}$, we obtain the$\pi_{1}(M)\underline{\rho_{\iota_{aut,\overline{D}}}}arrow SL_{n}(K_{y})arrow canonica1Aut(\mathcal{B}_{y})$on the Bruhat-Tits building$\mathcal{B}_{y}=\mathcal{B}_{n,K_{y}}$associated to the closed point

$y$

of

$\tilde{D}.$Similarly to the proof of Theorem 2.2, we may verify the following statement:$*9$

The fact that the natural action of $SL_{n}(K)$ on the Bruhat-Tits building
is type-preserving corresponds to theis without inversion. For the precise definition
fact that the natural action of $SL_{2}(K)$ on the Bruhat-Tits treeof type-preserving actions, refer to [Ga97, Section 52] for example.
$\mathcal{B}_{n_{\rangle}K}$$\mathcal{T}_{K}$69
Theorem 3.4 (H.-Kitayama). Let be a closed point ofand a lifl of torespectwithto . Consider the Bruhat-Tits building
associated to the closed
point ofand let be an element of. Then there exists a vertex of$\pi_{1}(M)$which is stabilised by under the action of(defined as above) if theon$evaluation-at-\gamma$ function
is regular at
In particular the action ofis nontrivial if ison the Bruhat-Tits building$an$ ideal pointof and is a lift of$\tilde{C}$

$x$

$x$

$y$

$\tilde{D}$$\tilde{\pi}$$\mathcal{B}_{y}$$\tilde{D}$$y$$\pi_{1}(M)$$\gamma$$\mathcal{B}_{y}$$\mathcal{B}_{y}$$\gamma$$x.$$I_{\gamma}$$\pi_{1}(M)$$\tilde{x}$$\mathcal{B}_{\overline{y}}$$\tilde{C}$$\tilde{x}.$$\tilde{y}$We remark that the converse of Theorem 3.4 does not hold in general for higherdimensional representations, unlike Theorem 2.2.
Construction of tribranched surfaces\S 3.3

We briefly summarise how to construct a tribranched surface from a nontrivial (andtype-preserving) action of
on the Bruhat-Tits building
associated to an ideal
point of the curve
obtained as in Theorem 3.4. The construction is similar to themethod of Stallings et al. which we have already explained in Section 1.
Let be a natural number greater than or equal to three, and assume that the boundaryof $M$ is not empty when is strictly greater than three. Let us take a triangulation of$M$ and lift it to the universal covering space
is equipped with theof $M$ . Therefore$\pi_{1}(M)$ -triangulation. Whenis strictly greater than three, we may take aof$M$because of the non-empty assumption on the boundary of . Due to the contractibilityof the Bruhat-Tits building, we may construct a-equivariant simplicial map(resp. from the 2-dimensional subcomplex
fromof
if is strictly greater than
three) to the 2-skeletonof . Composing it with a retraction
when isstrictly greater than three, we finally obtain a-equivalent piecewise-linear map$\pi_{1}(M)$$\mathcal{B}_{\overline{y}}$$\tilde{D}$

$n$

$n$

$\tilde{M}$$\tilde{M}$$spine^{*10}\tilde{K}$$n$$\tilde{M}$$\pi_{1}(M)$$\mathcal{B}_{\overline{y}}$$\tilde{M}$$\tilde{K}$$\tilde{M}$$n$$\tilde{M}arrow\tilde{K}$$\mathcal{B}_{\tilde{y}}^{(2)}$$\mathcal{B}_{\overline{y}}$$n$$\pi_{1}(M)$$\tilde{f}:\tilde{M}arrow \mathcal{B}_{\tilde{y}}^{(2)},$

which induces a piecewise-linear map
on the quotient complexes.
Since the target
of is now a simplicial complex ofdimension two, it is absurd to consider the set of all midpointsof edges.”’ We thus consider for a 2-dimensional simplicial complexof the barycentric
the 1-dimensional subcomplex
subdivision of consisting of the barycentres of all 1- and 2simplices and all the edges connecting them. The figure on theleft side shows
when consists of a single 2-simplex. We$f:Marrow \mathcal{B}_{\frac{(}{y}}^{2)}/\pi_{1}(M)$$\mathcal{B}_{\frac{(}{y}}^{2)}/\pi_{1}(M)$$\triangle$$f$$Y(\triangle)$$\triangle$$Y(\triangle)$$*10$$\triangle$

A spine of a topological manifold $M$ is a locally tamely embedded subcomplex $K$ of $M$ so that $K$ is a strong
deformation retract of $M$ and the natural inclusion $K\mapsto M$ is a simple homotopy equivalence.

70$\tilde{M}$$\pi_{1}(M)-equivariant\downarrow$$\downarrow$quotient

Figure 3. Construction of (essential) tribranched surfaces

deal withas a counterpart of the “set of all midpoints of edges”’ $E$ of thequotient graphin Section 1, and consider the inverse image ofunder the map(see Figure 3). We readily verify thatis a properly$M$
embedded tribranched surface in . As in Section 1, we may modify into an essential
tribranched surface by certain “local surgery.” For details refer to [HK14, Section 4.2].
Consequently we obtain the following theorem, which is the main result of [HK14]:$Y(\mathcal{B}_{\tilde{y}}^{(2)}/\pi_{1}(M))$$\Sigma’$$\mathcal{T}/\pi_{1}(M)$$f:Marrow \mathcal{B}_{\frac{(}{y}}^{2)}/\pi_{1}(M)$$Y(\mathcal{B}_{\frac{(}{y}}^{2)}/\pi_{1}(M))$$\Sigma’$$\Sigma’$$\Sigma$

Theorem 3.5 (H.-Kitayama). Let be a natural number greater than or equal tothree, and let $M$ be a 3-manifold. Furthermore assume that the boundary of $M$ is
not empty if is strictly greater than three. Then associated to each ideal pointof the $SL(n)$ -character variety’ we may construct an essential
$n$

$n$

$\tilde{x}$$X(\pi_{1}(M))_{SL(n)}/\mathbb{C}$tmbranched surface$\Sigma_{\overline{x}}$contained in $M.$

Conversely, we may also associate to an essential tribranched surface
a 2-complex
of groups in a way similar to the construction of the graph of groups associatedto an essential surface (the details shall appear in [HK14]; refer also to the final partof [PLDT14, Section 10 These results reveal a close relation between the concepts ofessential tribranched surfaces and (nontrivial) actions of 3-dimensional groups on certain$\Sigma$$\mathcal{G}_{\Sigma}$$\mathcal{G}s$

$S$

71
simply-connected 2-dimensional complexes:

Essential

Nontrivial actions of $\pi_{1}(M)$tribranched surfaces$clos\underline{erela}$tion

on certain simply-connected2-dimensional complexes(topological concept)\S 4(algebraic group action)

Several problems from the arithmetic viewpoint

In this final section we propose several problems and questions concerning actions of
3-manifold groups from the arithmetic viewpoint. The readers are referred to [PLDT14,
Section 10] for problems and questions from the topological viewpoint.
\S 4.1

From the arithmetic topological viewpoint

Arithmetic topology is a research field where one pursues analogous theory and analogous
phenomena between low-dimensional topology and number theory. Recently there has been
drastic progress in this research field by virtue of great eﬀorts of many people including
Masanori Morishita and his colleagues. See also Mizusawa’s article [Mi14] in this volume
for another aspect of the progress of arithmetic topology.
Table 2 is a part of the (what is called) (dictionary” of arithmetic topology, whichindicates correspondences between analogous objects in low-dimensional topology andnumber theory. According to this dictionary, the counterparts of the $SL(2)$ -character varieties of 3-manifold groups (which are certain kinds of “moduli spaces” of representationsof 3-manifold groups) are the universal deformation spaces of the absolute Galois groups,
which were first introduced by Barry Mazur in [Ma89] and have played crucial roles inrecent arithmetic geometry.
Moreover it has been widely known that there exist very
interesting (rigid analytic) curves in the universal deformation space of (certain) residual
Galois representations of the absolute Galois groupof the rational number field :namely curves of families of -adic modular for,$ms$ (Hida family, Coleman family, andultimately the “eigencurve”).
$*11$$G_{\mathbb{Q}}$$\mathbb{Q}$$p$

Question 4.1. For such a curve of a family of -adic modular forms, could we play thesame (or at least analogous) game as Culler and Shalen’s $method’$?
$p$Note, however, that there exist several crucial diﬃculties in this setting. Firstly, curvesof families of -adic modular forms are not algebraic curves but rigid analytic curves,
$p$

$*11$

For example, the deep study of the universal deformation spaces of Galois representations is one of the keys ofAndrew Wiles’s solution of Fermat’s Last Theorem.

72

Table 2. The (dictionary of arithmetic topology

and thus we could not apply results on (classical) birational geometry of algebraic curves
to them and need to heavily utilise rigid analytic geometry. Secondly, although we mayinterpret a point of rigid analytic curves as a valuation of a certain Banach algebra, it is
not a discrete valuation in general. Hence the associated Bruhat-Tits building no longer
admits a structure as a simplicial complex. Nevertheless there exists a possibility for us
to obtain a certain action of the absolute Galois group on (combinatorially complicated)
object, and it might be interesting to study such a mysterious action of the absolute Galois
representation.
$a$\S 4.2

From the zeta functional viewpoint

As we have also seen in Section 1, we may associate to an essential surface containedin a canonical manner. Moreover the underlying
in a 3-manifold $M$ a graph of groups
is a finite graph due to the compactness of $M$ . For a general finite graph
graph of,we may consider the zeta function of the graph (or the Ihara zeta function of )which has its origin in Ihara’s thorough study of a -adic analogue of the Selberg zeta
$S$$\mathcal{G}s$$\mathcal{G},$$\mathcal{G}s$$\mathcal{G}$$\mathcal{G}$$p$$\zeta_{\mathcal{G}}(u)$73functions in [Iha66] and has been developed by many people including Toshikazu Sunada,Ki-ichiro Hashimoto, Hyman Bass and Audrey Terras.
Under the background above let us consider the zeta function of the graph (of groups)
associated to an essential surface , which seems to contain certain information of
the essential surface . Moreover if the essential surface is associated to an ideal pointof (a curve
contained in) the $SL(2)$ -character variety $X(\pi_{1}(M))_{SL(2)}$ , we may obtain
another zeta function; namely the Hasse-Weil zeta function associated to $X(\pi_{1}(M))_{SL(2)}$(or associated to ). See also Harada’s article [Harada14] in this volume for the HasseWeil zeta functions of $SL(2)$ -character varieties. As Ihara has already observed in [Iha68],
the zeta functions of certain graphs might have close relations with the Hasse-Weil zetafunctions of certain algebraic curves, and thus it might be interesting to investigate therelation between the zeta functions of the graphs associated to essential surfaces and theHasse-Weil zeta functions of character varieties.
We finally give one remark. When we consider the zeta function of , we ignore theinformation of vertex and edge groups of , and the underlying graph (that is, the dualgraph of ) might become too simple. For example, let us consider the knot complementof an arbitrary knot $K$ . In this casecontains the Seifert surface
of $K$ which isessential. The graph of groups
associated to the Seifert surface
then consists of asingle vertex with the trivial vertex group and a single edge representing
Therefore
the zeta function
reduces to an uninteresting function $(1-u)^{-1}$ because
admitsonly one prime cycle of length one.
This observation implies that we should impose
certain constraints on essential surfaces (or on 3-manifolds themselves) when we consider
the zeta functions of the graphs associated to them, or more ultimately, we should try
to construct more sophisticated zeta-function-like invariants of graphs of groups whichreflect the information of their vertex and edge groups.
$\mathcal{G}s$

$S$

$S$

$S$

$C$

$C$

$\mathcal{G}_{S}$$\mathcal{G}s$$S$$V_{K}$$V_{K}$$S_{K}$$S_{K}$$\mathcal{G}_{S_{K}}$$S_{K}^{*12}$$\zeta_{\mathcal{G}s_{K}}(u)$$\mathcal{G}_{S}$$*13$

References[PLDT14] Tomotada OHTSUKI edt., Problems on Low-dimensional Topology, 2014, RIMS
K\^oky\^uroku, this volume (2014).
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$*12$

According to the theory of graphs of groups, this implies the classical fact that the fundamental group of a knot
complement is represented as a HNN extension of that of its Seifert surface.
$*13$
Thus in many references one does not consider the zeta functions for such circuit graphs.

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75
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Department of Mathematics
School of Science and Technology for Future Life
Tokyo Denki University
5, Senju-Asahi-chi, Adachi-ku, Tokyo 120-8551

JAPAN-mail address: t-hara@mail.dendai.ac.jp$E$$\ovalbox{\tt\small REJECT})\ovalbox{\tt\small REJECT} ffl\star 3\cdot**\gamma_{\backslash }\}3_{\lceil\beta\Re\ovalbox{\tt\small REJECT}^{\tau}}^{r_{n}}*F^{1\rfloor}$

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