The Geometry of The Word Problem in 3-Manifold
Groups
Tim Susse
University of Nebraska - Lincoln
April 17, 2016
Joint work with Mark Brittenham and Susan Hermiller
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Groups and the Word Problem
Let G = hA | Ri be a finite presentation of a group.
The word problem for G asks if there is an algorithm that determines
where a word w ∈ A∗ is trivial in G.
Not every group has a decidable word problem (Boone & Novikov), but
many do.
Hyperbolic groups have word problem solvable by Dehn’s
algorithm
Nilpotent and Solvable groups
3-manifold groups (as a consequence of geometrization)
Of particular interest are uniform algorithms over naturally defined
classes and bounds on computational complexity (e.g. automatic
groups)
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(Auto)stackable groups
Let G = hAi be a finitely generated group, ΓA (G) its Cayley graph.
Definition
A finitely generated group is called stackable over a generating set A if
ΓA (G) has a spanning tree T and there is a flow function Φ defined on
the edges of ΓA (G) so that:
Φ(e) is a path in ΓA (G) between the initial and terminal endpoints
of e and Φ(e) = e for all e ∈ T
there are no infinite sequences {ei } of edges so that ei is an edge
on the path Φ(ei−1 ).
there exists k so that the length of Φ(e) is at most k.
Such a G is called autostackable if the flow function is computable by a
finite state automaton.
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Autostacking Z2
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Autostacking Z2
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Autostacking Z2
Figure: Φ(e) depends only on the label of e and the last letter of the normal
form of the initial vertex
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Properties
Stackable groups are finitely presented
Autostackable groups have solvable word problem (using only
finite state automata)
There exist stackable groups with unsolvable word problem
(Hermiller, Martı́nez-Perez, S.)
Stackability is a priori generating set dependant
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Properties
Stackable groups are finitely presented
Autostackable groups have solvable word problem (using only
finite state automata)
There exist stackable groups with unsolvable word problem
(Hermiller, Martı́nez-Perez, S.)
Stackability is a priori generating set dependant
Groups with (asynchronously) automatic structures with prefix
closed unique normal forms are autostackable
(Brittenham-Hermiller-Holt)
Groups admitting finite complete rewriting systems are
autostackable(BHH)
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3-Manifolds
Let M = M 3 be a closed orientable 3-manifold.
Prime Decomposition
M can be decomposed as a connect sum M = M1 #M2 # · · · #Mk , with
each Mi a prime manifold, in an essentially unique way
JSJ Decomposition
A prime 3-manifold M has a unique (up to isotopy) collection of
π1 -embedded tori which decompose M into three-manifolds with
incompressible torus boundary.
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3-Manifolds
Let M = M 3 be a closed orientable 3-manifold.
JSJ Decomposition
A prime 3-manifold M has a unique (up to isotopy) collection of
π1 -embedded tori which decompose M into three-manifolds with
incompressible torus boundary.
Geometrization
Every prime 3-manifold either:
admits a metric based on one of Thurston’s 8 geometries
g2 , Nil, Sol)
(S 3 , E3 , H3 , S 2 × R, H2 × R, SL
admits a non-trivial JSJ decomposition so that each of the
complementary components is either Seifert fibred or atoroidal
(and so admits a finite volute hyperbolic metric)
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Seifert Fibred Pieces
Figure: Glue the top to the bottom by a rotation of
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pπ
.
q
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Seifert Fibred Pieces
Figure: The base orbifold. M is a circle bundle over B
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Knots and Hyperbolic Pieces
(a) The Trefoil
(b) Figure-8
(c) Borromean Rings
Figure: The complement of these links in S 3 are 3-manifolds with
incompressible torus boundary. Two are hyperbolic, one is Seifert fibred.
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Knots and Hyperbolic Pieces
(a) The Trefoil
(b) Figure-8
(c) Borromean Rings
Figure: The complement of these links in S 3 are 3-manifolds with
incompressible torus boundary. Two are hyperbolic, one is Seifert fibred.
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Automaticity & Other Attempts
Theorem (Epstein-Canon-Holt-Levy-Patterson-Thurston, 1992)
Let M be a 3-manifold whose prime decomposition contains no Nil or
Sol pieces, then π1 (M) is automatic.
Theorem (Thurston (1992), Brady (2001))
The fundamental group of a closed Nil or Sol manifold is not
asynchronously automatic.
Theorem (Bridson-Gilman, 1996)
Every 3-manifold groups admits an asynchronous combing by an
indexed language. This, however, cannot be improved to regular or
even context-free.
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Results
Theorem (Brittenham-Hermiller-Holt, 2014)
If M is a 3-manifold all of whose prime factors are geometric, then
π1 (M) is autostackable.
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Results
Main Theorem (Brittenham-Hermiller-S.)
Let M be a 3-manifold with incompressible torus boundary. Then
π1 (M) is autostackable.
Theorem (Brittenham-Hermiller-S.)
Let G be hyperbolic relative to a collection H of virtually abelian
subgroups. Then G is autostackable and further, for each H ∈ H there
is an autostackable structure which determines whether a word in the
generators of G represents an element of H.
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Autostackability Respecting a Subgroup
Definition
Given a finitely generated subgroup H ≤ G, we say that G is
autostackable respecting H if G has an autostackable structure over a
generating set A (containing generators B for H) so that the spanning
tree T for G is formed from a spanning tree T 0 for H, along with the
H-orbit of a transversal tree T 00 .
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Autostackability Respecting a Subgroup
Definition
Given a finitely generated subgroup H ≤ G, we say that G is
autostackable respecting H if G has an autostackable structure over a
generating set A (containing generators B for H) so that the spanning
tree T for G is formed from a spanning tree T 0 for H, along with the
H-orbit of a transversal tree T 00 .
Equivalently, there is a (right) transversal T for H in G so that G has a
set of normal forms that look like normal forms for H followed by the
normal form for an element of T (N = NH NT ).
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Autostackability Respecting H
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Autostackability Respecting H
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Autostackability Respecting H
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Autostackability Respecting H
Such an autostackable structure solves the membership problem for
H, since every element of H has a normal form in the tree T 0 .
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Gluing Together Graphs of Groups
Figure: π1 (M) is obtained from π1 (N) and π1 (N 0 ) by a graph of groups
construction.
We can do this with more manifolds and obtain any possible dual
graph.
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Autostacking Graphs of Groups
Theorem (BHS)
Let M be a 3-manifold with each incompressible torus boundary.
Suppose that for each piece Ni of the JSJ decomposition of M and
each choice of component C of ∂Ni , π1 (Ni ) has an autostackable
structure respecting π1 (C). Then π1 (N) is autostackable.
To use this theorem in our case, we need a geometric condition to
ensure that every piece of JSJ decomposition is autostackable
respecting any choice of boundary component. The property we need
relies on negative curvature of each of the JSJ pieces (called coset
fellow travelling).
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JSJ Pieces
Proposition (BHS)
If N is a finite volume geometric 3-manifold with incompressible torus
boundary, then π1 (N) is autostackable respecting any choice of
representative and component of ∂N.
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JSJ Pieces
Proposition (BHS)
If N is a finite volume geometric 3-manifold with incompressible torus
boundary, then π1 (N) is autostackable respecting any choice of
representative and component of ∂N.
Corollary (BHS)
Let M be a 3-manifold with (possibly empty) incompressible torus
boundary, then π1 (M) is autostackable.
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Nice Properties
For M geometric with torus boundary, the autostacking on π1 (M) has
some nice properties:
when M is hyperbolic, the generators are chosen so that the
peripheral subgroups are isometrically embedded
every normal form is a (1,k)-quasigeodesic, with k depending only
on M
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Nice Properties
For M geometric with torus boundary, the autostacking on π1 (M) has
some nice properties:
when M is hyperbolic, the generators are chosen so that the
peripheral subgroups are isometrically embedded
every normal form is a (1,k)-quasigeodesic, with k depending only
on M
Question
If M is prime and non-geometric, can the normal forms for π1 (M) be
chosen to be quasigeodesic? Does this give rise to an automatic
structure on π1 (M) with unique, prefix closed normal forms? Is π1 (M)
shortlex automatic? shortlex autostackable?
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Thank you
Thank You
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