几何 群论 讲义

A Short course in geometric group theory

Notes for the ANU Workshop January/February1996

Walter D.Neumann and Michael Shapiro

2Geometric Group Theory

2Elementary properties of hyperbolic groups

1)Behaviour of geodesics and quasi-geodesics:progression in geodesic corridors.

2)Hyperbolicity is a quasi-isometry invariant.

3)Finite presentation,solubility of the word and conjugacy problems.

4)Finitely many conjugacy classes of torsion elements.

5)The Rips complex and.

6)The boundary.

3Isoperimetric inequalities

1)Area of a word via products of conjugates of relators.

2)Area of a word via V an Kampen diagrams.

3)Dehn’s functions and isoperimetric functions.

4)A group has soluble word problem if and only if it has a recursive Dehn’s function if

and only if it has a sub-recursive Dehn’s function.

5)Equivalence relation and ordering for isoperimetric functions.

6)Consequently solubility of the word problem is a geometric property.

4The JSJ decomposition

1)Splittings of hyperbolic groups.

5Regular languages,automatic,bi-automatic and asynchronously automatic groups

1)De?nitions.

2)The fellow traveler property.

3)The seashell:?nite presentation,quadratic isoperimetric inequality and quadratic time

word problem.

4)Closure properties.

i)Free products.

ii)Direct products.

iii)Free factors.

iv)Finite index subgroups.

v)Finite index supergroups.

vi)HNN extensions and free products with amalgamation over?nite subgroups. vii)Bi-automatic groups are closed under central quotients and direct factors.

5)Famous classes of automatic groups.

Geometric Group Theory3

i)Hyperbolic groups including:free groups,?nite groups,most small cancella-

tion groups,fundamental groups of closed negatively curved manifolds and other negatively curved spaces.

ii)Small cancellation groups.

iii)Fundamental groups of geometric three manifolds except for those containing a nil or solv manifold as a connected sum component.

iv)Coxeter groups.

v)Mapping class groups.

vi)Braid groups and more generally,Artin groups of?nite type.

vii)Central extensions of hyperbolic groups.

viii)Many amalgams of hyperbolic groups along rational subgroups.

ix)Many groups that act on af?ne buildings.

6)Regular languages,cone types and the falsi?cation by fellow traveler property.

7)Bi-automatic groups

i)Hyperbolic groups are geodesically bi-automatic.

ii)Bi-automatic groups have soluble conjugacy problem.

8)Asynchronous automaticity

6Equivalence classes of automatic structures

1)The asynchronous fellow traveller property,rationality and bi-automaticity.

7Subgroups of automatic groups

1)Let be a(synchronous or asynchronous)automatic structure for,and suppose

.Then the following are equivalent:

i)is-rational.

ii)is-quasi-convex.

iii)In the case where is hyperbolic and is synchronous,these are equivalent to: is quasi-geodesic in.

2)A rational subgroup of an automatic(bi-automatic,asynchronously automatic,hyper-

bolic)group is automatic(bi-automatic,asynchronously automatic,hyperbolic).

3)Centers and centralizers of bi-automatic groups are rational,hence bi-automatic.

i)Consequently a hyperbolic group does not contain a subgroup.

8Almost convexity

1)De?nition.

2)Building the Cayley graph.

3)Almost convexity is not a group property.

4Geometric Group Theory

9Growth functions and growth rates

1)Finite cone types implies rational growth function.

2)Gromov’s theorem:polynomial growth implies virtually nilpotent.

Geometric Group Theory5 0.Introduction

Geometric group theory is the study of groups from a geometric viewpoint.Much of the essence of modern geometric group theory can be motivated by a revisitation of Dehn’s three decision-theoretic questions,which we discuss below,in light of a modern viewpoint.This viewpoint is that groups may be pro?tably studied as geometric objects in their own right.This connection between algorithmic questions and geometry is at?rst surprising,and is part of what makes the subject attractive.

As Milnor’s theorem(see below)teaches us,the geometry exists both in the group itself and in the spaces it acts on.This is another powerful motivation for the subject,and was historically one of the turning points.

0.1–4Dehn’s problems.

Early in this century Dehn proposed three seminal questions[D].

Suppose we have a group given in some way,and a set of generators for the group.The word problem asks if there is a procedure to determine whether two words and in these generators represent the same element of.Since we may look at,it is equivalent to ask for a procedure to decide if a word represents the identity.

Notice that if and are two?nite generating sets for then the word problem is soluble with respect to if and only if it is soluble with respect to.For suppose we are given a word in the letters of.Each letter has the same value in as some word in the letters of.Thus it requires only a?nite look-up table to translate a word in the letters of into one in.

However,there is less here than meets the eye.We have said nothing about how to ?nd the?nite look-up table.Thus while we have demonstrated the existence of an algorithm for the word problem in from the existence of one in,we may have no way of?nding that solution.

The conjugacy problem asks if there is a procedure to determine whether the elements and of represented by two words and are conjugate in,that is,does there exist with.Note that if such a procedure exists,then,taking ,we solve the word problem also.

The isomorphism problem asks for a procedure to determine whether two groups are isomorphic.The groups are usually assumed to be given by presentations(a“presentation" is a collection of generators for the group plus some suf?cient collection of relations among these generators).

Dehn’s three questions are remarkable in that they precede by many years the for-malization of“procedure"as“algorithm"by Church,Turing,et al.and by two more decades the resolution of the word problem by Boone and Novikov in the1950’s.They show that there is a?nitely presented group with recursively unsolvable word problem. The unsolvability of the conjugacy problem is immediate,and that of the isomorphism problem also follows.For more details see https://www.sodocs.net/doc/7d16843848.html,ler’s notes for this Workshop.

6Geometric Group Theory

0.5Cayley graphs

There is a standard method for turning a group and a set of generators into a geometric object.Given the group and generating set we produce the directed labeled graph .The vertices of are the elements of and we draw a directed edge from to with label for each and.We will refer to this edge as.

This graph is called the Cayley graph.It comes equipped with a natural left action of .acts on vertices by left-multiplication and carries the edge to .

By forgetting directions on edges and making each edge have length,we can turn into a metric space.This induces the word metric on the vertex set:

in

This metric depends of course on the chosen set of generators.But we will see below that up to“quasi-isometry”it does not depend on this choice(if the generating set is?nite) and that many geometric properties depend only on.

We use to denote the set of all words on letters of,including the empty word and for we use

for.The path is a geodesic if

.Evidently,a word represents the identity if and only if it is a closed loop.It follows that Theorem.has soluble word problem if and only if there is an algorithm capable of constructing any?nite portion of.

For suppose we are in possession of such an algorithm,and we are given the word. The path lies entirely inside the ball of radius around the identity.We use our algorithm to construct this ball and then follow to see if it returns to the identity.

On the other hand,suppose we are given an algorithm to solve the word problem in .To construct the ball of radius around the identity,we enumerate the words of length less than or equal to in.We then use our algorithm for the word problem to determine which of these are equal in.Equality in is an equivalence relation on .Pick a representative(say a shortest representative)in each equivalence class.Now for each and each pair of representatives,use word problem algorithm to determine if is an edge.

Geometric Group Theory7 0.7Dehn’s solution to the word problem for hyperbolic surface

groups.

The previous algorithms are wildly impractical in most situations.There is a beautiful and highly ef?cient solution to the word problem for hyperbolic surface groups.Let us start by describing an inef?cient solution to the word problem in.

If we take in the standard presentation,the Cayley graph embeds in the Euclidean plane as the edges of the tessellation by squares.We can see this embedding as an expression of the fact that acts by isometries on the Euclidean plane.The quotient of the Euclidean plane by this action is the torus.Now we are entitled to see the tessellation of the plane by squares as being the decomposition of the plane into copies of a fundamental domain for this action.However,there is a little piece of sleight-of-hand going on here.To see this,let’s start with the torus.

If we cut the torus open along two curves,it becomes a disk,in fact,if we choose the two curves correctly,it is a square.That is,the torus is the square identi?ed along its edges.If we look for the generators of the fundamental group,they are dual to the curves we have cut along.That is,if we take a base point in the middle of the square,is the curve which starts at the base point,heads towards the right edge of the square,reappears at the left edge and continues back to the base point.Likewise,is the vertical path which rises to the top edge and reappears at the bottom.Thus the fundamental group is generated by the act of crossing either of the cut curves.

Thus,after choosing a base point,the natural relation between the tessellation of the plane and the embedding of the Cayley graph into the plane is that they are dual.That is,there is a vertex of the Cayley graph in the center of each copy of the fundamental domain,and there is an edge of the Cayley graph crossing each edge of the tessellation.

The reason it was not immediately obvious that the tessellation and the Cayley graph are two different things is that the tessellation of the plane by squares is self-dual.That is,if we start with the tessellation of the planes by squares and replace each vertex by a2-cell and each2-cell by a vertex(thus getting new edges crossing our old edges)the result is another tessellation of the plane by squares.Contrast this with the tessellation of the plane by equilateral triangles which is dual to the tessellation of the plane by regular hexagons.

Now as we have seen,?nding the Cayley graph for solves the word problem for ,but it does not lead to a particularly ef?cient solution.

This situation changes if we turn to the fundamental group of a hyperbolic surface, i.e.,a surface of genus2or more.If we wish to cut open a surface of genus2,we will need4curves,and when we have cut it open,we will have an octagon as our fundamental domain.Likewise,looking at the vertex where our cut curves meet shows that we will want to tessellate something with8octagons around each vertex.This suggests that we would like regular octagons with interior angles of.We can have this if we choose to work in the hyperbolic plane.In fact,the hyperbolic plane can be tessellated by regular octagons with interior angles of.

Once again,the Cayley graph is dual to the tessellation,and this tessellation is also self-dual,so the Cayley graph embeds as the edges of this tessellation.In fact,reading

8Geometric Group Theory

the labels on an octagon tells us that our fundamental group has the presentation Given a sub-complex of this tessellation,we de?ne to be together with any fundamental domains which meet.Now suppose we start with a vertex of this tessellation(identi?ed with the identity)and consider for successive values of.We can then observe that each fundamental domain that meets the boundary of meets it in at least of its edges.

Suppose now that we are given a word which represents the identity.We consider as a path based at.There is a smallest so that lies entirely in.If then is the empty word.If,then there is some portion of which lies in,but not in.Suppose is a maximal such portion.Then one of2things happens.Either:

1)returns to along the same vertex by which it left it,in which case

is not reduced or

2)travels along the edges of some2-cell which meet the boundary of

.Call this portion.

In the?rst case,can be reduced in length by deleting a subword of the form, producing a new word with the same value in.

In the second case,we take to be the path so that forms the boundary of. We now have evaluating to the identity,so and have the same value in. Further,.

If we work with a surface of genus greater than,the numbers change,but the argument stays the same.Thus we have proven:

Theorem(Dehn).Let be the fundamental group of a closed hyperbolic surface.Then there is a?nite set of words with each evaluating to the identity, so that if is a word representing the identity,then there is some so that and.

This gives a very ef?cient solution to the word problem.Given a word we look for an opportunity to shorten it and do so if we can.After at most such moves we either arrive at the empty word or we have no further opportunities to shorten our word. If the?rst happens,we have shown that represents the identity.If the second happens, we have shown that it does not.

1.1Hyperbolicity

Let be a?nitely generated group with?nite generating set and let be the corresponding Cayley graph.

i).We say that has thin triangles if there exists a such that if,,and are sides of a geodesic triangle in then lies in a-neighbourhood of.

ii).We can give a parameterized version of the above condition.We?rst note the following.If is a geodesic triangle,then the sides of decompose(as paths parameterized by arclength)as,,so that

Geometric Group Theory9 ,,and.(This is the triangle inequality.Exercise!)

The parameterized version of the previous condition is that there exists such that for any such triangle each of the pairs,;,;,-fellow travel,that is for and so on.The reader may wish to check that suf?ces.

iii).We say that has thin bigons if there is a such that for every pair of geodesics and with the same endpoints,for.(We call such a pair a geodesic bigon.)Warning:we are here treating as a bona?de geodesic metric space.Accordingly,we must allow the endpoints of geodesics to occur in the interior of edges.

iv).We shall see that a group obeying any of these conditions is?nitely presented.That is to say,there is a presentation

where consists of a?nite set of words on the set of generators and

is isomorphic to the quotient of the free group on by the normal closure of.

Another way of saying this is that a word in the letters of represents the identity in if and only if it is freely equal to a product of conjugates of elements of.Thus a word in represents the identity if is equal(in the free group on to an expression of the form

We say that has a linear isoperimetric inequality if there is a constant so that for any trivial word we can satisfy this equation with.

v).We say that has a sub-quadratic isoperimetric inequality if there is a sub-quadratic function so that for any word presenting the identity,can be freely expressed as the product of at most conjugates of de?ning relators.

vi).We say that has a Dehn’s algorithm if there is a?nite set of words with each evaluating to the identity,so that if is a word representing the identity,then there is some so that and.

10Geometric Group Theory

We have seen that this gives an algorithm.Given a word,we can replace it with the shorter word which evaluates to the same group element.If does in fact represent the identity,after at most such moves,we are left with the empty word. If does not represent the identity,after at most such moves,we are left with a word which we cannot shorten.

vii).We will say that geodesics diverge exponentially in if there is a constant and an exponential function()with the following property:Suppose that and are geodesic rays based at a common point.Suppose that there is a value so that .Suppose now that is a path connecting to

and that lies outside the ball of radius around.Then. viii).We will say that geodesics diverge uniformly in if there is a constant and a function,,with the following property:Suppose that and are geodesic rays based at a common point.Suppose that there is a value so that .Suppose now that is a path connecting to

and that lies outside the ball of radius around.Then. Theorem.All the above conditions are equivalent and independent of generating set.A group satisfying them is called word hyperbolic.

Conditions i),ii),and vii)are equivalent to each other for any geodesic metric space and are characterizations of a type of hyperbolicity in such spaces called Gromov-Rips hyperbolicity.In a class of spaces including complete simply connected riemannian manifolds they are also equivalent to linear isoperimetric inequality.In these spaces linear isoperimetric inequality means the existence of a constant such that a closed loop can always be spanned by a disk of area at most.

The thin bigons condition is equivalent to Gromov-Rips hyperbolicity in any graph with edges of unit length(see[Pa2]),but certainly not in arbitrary geodesic metric spaces (think of euclidean space!).

Conditions i),ii),iv),vi),and vii)can be found in general expositions such as[ABC], [B],[C3],[CDP],[GH].Conditions v)and viii)can be found in[Pa1].

1.2Cayley graphs and group actions

We say that a metric space is a geodesic metric space if distances in are realized by geodesics in That is,given,there is a path connecting and so that and is minimal among all paths connecting and.

A map is a quasi-isometric map if for all,

Geometric Group Theory11 Suppose now that is a geodesic metric space.Suppose that is a?nitely generated group which acts by isometries on.Suppose further that this action is discrete and co-compact.“Discrete”means that if is a sequence of distinct group elements then for the sequence does not converge(this is slightly weaker than“proper discontinuity”).“Co-compact"means that the orbit space is compact.

Let us?x a generating set for and pick a basepoint,.The map which takes to takes to,is-equivariant,and is?nite-to-one since the action of is discrete.For each,we choose a path from to.We now have a map

de?ned by taking each vertex of to,and each edge of to. (Strictly speaking,we must parameterize both the edge and the path by the unit interval and take to.)

Theorem(Milnor)[M].The map is a quasi-isometry.

This is not a particularly dif?cult theorem—indeed,the reader may wish to try to prove it as an exercise.However,as mentioned in the Introduction,it was one of the turning points in the development of modern geometric group theory.

1.3Groups acting on hyperbolic spaces.

In view of Milnor’s Theorem,we will want to see that any space which is quasi-isometric to a hyperbolic space is itself hyperbolic.This will show that co-compact discrete action of a group on a hyperbolic space“transfers"the hyperbolicity of to.It will also show that hyperbolicity is independent of generating set.

2.1Quasi-geodesics in a hyperbolic space

A geodesic is an isometric map of the interval.(Y ou may take this a de?nition of.)A-quasi-geodesic is a-quasi-isometry of the interval .Here are several characterizations of the relationship between geodesics and quasi-geodesics in a hyperbolic space.

Given a path it is often convenient to extend it to a map de?ned on by setting for.We use this convention in the following theorem. Theorem.There are,,and so that if

is a-quasi-geodesic and is a geodesic in and is-hyperbolic,and and have the same endpoints,then each of the following hold:

i).Each of and is contained in an-neighbourhood of the other.

ii).and asynchronously-fellow travel.That is,there is a monotone surjective reparameterization of so that for all we have. iii).progresses at some minimum rate along.That is,the reparameterization of ii) can be chosen so that implies.

12Geometric Group Theory

This last is sometimes referred to as “progression in geodesic corridors."

2.2Hyperbolicity is a quasi-isometry invariant

We now prove that hyperbolicity is a quasi-isometry invariant using the previous theorem and a picture.

Theorem.Suppose that

and are quasi-isometric geodesic metric spaces.If is

hyperbolic,so is

.

2.3Word and conjugacy problem for hyperbolic groups

It follows immediately from the existence of a Dehn’s algorithm that a hyperbolic group has a highly ef?cient solution to its word problem.Most early solutions to the conjugacy problem used the boundary of a hyperbolic group,which will be described later.We will give a proof in 5.7due to Gersten and Short which works in the more general setting of biautomatic groups,and which does not use the boundary.

2.4Torsion in hyperbolic groups

There is a charming proof that hyperbolic groups have ?nitely many conjugacy classes of torsion elements based on the Dehn’s algorithm.See for example,[ABC ].

2.5The Rips Complex

A fundamentally important property of a hyperbolic group is:

Theorem (Rips).Let be a hyperbolic group.Then acts properly discontinuously on a ?nite dimensional contractible complex with compact quotient.

Such a complex is easy to describe.We start with a Cayley graph for and take the geometric realization of the following abstract simplicial complex

Geometric Group Theory13 for an appropriate bound.V ertex set of is and the simplices consist of subsets of of diameter at most in the word metric.It turns out that

always suf?ces.A very ef?cient proof of Rips’theorem is given in[ABC].

This theorem has important homological implications for.For example,if is virtually torsion free(i.e.,has a torsion free subgroup of?nite index)then it has?nite virtual cohomological dimension and in any case its rational homology and cohomology is?nite dimensional.It is an important open problem whether a hyperbolic group is always virtually torsion free.

Another interesting consequence is that non-vanishing homology implies a lower bound for the hyperbolicity constant for among all?nite generating sets for .

2.6The boundary of a hyperbolic group

For a hyperbolic group(in fact more generally for any Gromov/Rips hyperbolic metric space)there is a natural compacti?cation of by adding a“boundary at in?nity.”Roughly speaking the boundary consists of the set of all ways to travel off to in?nity. One way of making this precise is to de?ne a geodesic ray in a Cayley graph for as an isometry of into and to say two rays are equivalent if they fellow travel. The boundary(which in fact is a quasi-isometry invariant and thus does not depend on the choice of generating set)is the set of equivalence classes of geodesic rays.The topology on can be de?ned in a multitude of ways.Basicly two points are close if their rays fellow travel for a long time.One way to formalize this is to take the compact open topology on the set of rays and take the quotient topology on.

There are also many ways of describing how to attach this boundary to or its Cayley graph.One obtains a compact Hausdorff space into which both and embed. We leave as an exercise how to attach the boundary:the construction is highly stable in the sense that any reasonable answer you give will be correct(see[NS1]).

3.1Area of a word that represents the identity

As we have seen,a group with presentation

is isomorphic to,where is the free group on and is the normal closure of in.It follows that a word represents the identity in if and only if it is freely equal(that is,equal in)to an expression of the form

where each.Thus,solving the word problem in means determining the existence or non-existence of such an expression for each.

A naive approach would be to start enumerating all such expressions in hopes of ?nding one freely equal to our given.If represents the identity,we must eventually

14Geometric Group Theory

?nd the expression that proves this.The problem with this approach is knowing when to give up if does not represent the identity.Let us formalize this quandary.

Suppose we take a word representing the identity in.We de?ne the area,

to be the minimum in any such expression for.

3.2Van Kampen diagrams

The choice of the word area is motivated the notion of a V an Kampen diagram for. Such a diagram is a labeled,simply connected sub-complex of the plane.Each edge of is oriented and labeled by an element of.Reading the labels on the boundary of each2-cell of gives an element of.is a V an Kampen diagram for if

.

reading the labels around the boundary of gives

Theorem.Each Van Kampen diagram with2-cells for gives a way of expressing as a product.Each product for gives a Van Kampen diagram for with at most2-cells.In particular,has a Van Kampen diagram if and only if represents the identity.

Thus,we could have de?ned to be the minimum number of2-cells in a V an

Kampen diagram for.

3.3Dehn’s function of a presentation

We de?ne the Dehn’s function of the presentation to be

This maximum is taken over words presenting the identity.We say is an isoperimetric

function for if.(As we shall see below,“isoperimetric function”

is often used in a sense between these two.)

Geometric Group Theory15

3.4Isoperimetric functions and the word problem

Either of these functions answers the question,“When should we give up?" Theorem.has a soluble word problem if and only if it has a recursive Dehn’s function if and only if it has a sub-recursive Dehn’s function(and the same for isoperimetric functions).

A function is recursive if it can be computed by some computer program.It is sub-recursive if there is a recursive function so that for all. Proof.If a given word represents the identity,we can eventually exhibit a V an Kampen diagram for it by sheer perseverance.The Dehn’s function(if we can compute it or an upper bound for it)tells us the maximum number of2-cells in any V an Kampen diagram we must consider.Now,the number of possible V an Kampen diagrams with a given number of two cells is unbounded,because such diagrams may have long1-dimensional portions.However,the length of the boundary of such a diagram is at least twice the total length of the1-dimensional portions.Thus any possible diagram for has at most 2-cells and at most

16Geometric Group Theory

3.6The word problem is geometric

Theorem.The property of having soluble word problem is a quasi-isometry invariant of ?nitely presented groups.

This follows because the property of being sub-recursive is an equivalence class invariant of functions.

Since groups with unsolvable word problem exist,isoperimetric functions can be of enormously rapid growth.In fact,by their de?nition,non-sub-recursive functions can be thought to have“inconceivably rapid growth.”

It is not hard to?nd groups with solvable word problem with very fast growing Dehn functions—for example,Gersten has pointed out[G]that for any the function (levels of exponent)is a lower bound for the isoperimetric function for the group(the notation is a shorthand for).Clearly,

the algorithm proposed above for the word problem—enumerating all V an Kampen diagrams up to the size given by the Dehn function—is absurd when one has a Dehn function that grows this fast(or even exponentially fast).For speci?c groups much faster algorithms can often be found.

4JSJ decomposition.

The name JSJ refers to Johannson,and Jaco and Shalen,who developed a theory,building on earlier ideas of Waldhausen,for cutting irreducible three-dimensional manifolds into pieces along tori and https://www.sodocs.net/doc/7d16843848.html,ter Thurston explained these decompositions from a geometric point of view in his famous“Geometrization Conjecture"for3-manifolds.One can describe JSJ decomposition in terms of amalgamated product and HNN decomposition of the relevant fundamental group,and in fact a purely group theoretic version of the theory has been worked out by Kropholler and Roller.

Around1992Zlil Sela pointed out that analogous decompositions appear to exist for groups in a much broader range of situations.First Sela did this for torsion free hyperbolic groups and then Sela and Rips extended it to general torsion free?nitely presented groups.This is currently a very active area of research,and we can only touch on some of its coarsest elements here.Since some of the major players are here at this workshop(Swarup,Bowditch),we can hope to learn more in seminars.In fact we are indebted to them for the information in this section,although any errors in it are our responsibility.

The theory also has origins in work of Stallings on“ends of groups."The set of ends of a locally compact space is the limit over larger and larger compact subsets of of the set of components of.The set of ends of a?nitely generated group is the set of ends of any connected space on which acts freely with compact quotient —for example one may take a Cayley graph of the group.The de?nition turns out to be independent of choices.For example,has two ends,has one end for,and a free group on two or more generators has in?nitely many ends,since its Cayley graph

Geometric Group Theory17

is an in?nitely branching tree.For a hyperbolic group it is known that the set of ends coincides with the set of components of the boundary of the group.

Stallings showed that a?nitely generated group can have only(if the group is ?nite),,,or in?nitely many ends.Moreover,if ends then the group is virtually in?nite cyclic and if in?nitely many then the group can be“split"along a?nite subgroup, that is,it is an amalgamated free product or HNN extension amalgamated along a?nite subgroup.In the latter case of“decomposing along a?nite subgroup,”one might try to iterate the decomposition if the component groups that are being amalgamated still have in?nitely many ends.Work of Dunwoody[Du]shows that for a?nitely presented group this iteration eventually ends.One codes the result in what is called a“graph of groups." This is a?nite graph with groups assigned to vertices of the graph and subgroups of the vertex groups assigned to adjacent edges of the graph as a scheme for describing how to repeatedly amalgamate the vertex groups along the edge groups using amalgamated free products or HNN extensions.In our case all the edge groups are?nite and the vertex groups each have at most one end,since they cannot be further decomposed.This is, as it were,the?rst stage of JSJ decomposition,and leaves us with one-ended groups to decompose.

We now restrict to one-ended hyperbolic groups.In this case JSJ decomposition concerns splitting along virtually in?nite cyclic subgroups.The main question is when such splittings exist.We have:

Theorem(Paulin and Rips).If is a one ended hyperbolic group with

then splits along a virtually in?nite cyclic group.

By we mean the outer automorphism group:the quotient of by the group of inner automorphisms,that is automorphisms induced by conjugation by a group element.

Theorem(Swarup and Scott,Bowditch).If is a hyperbolic group with one end and is a virtually in?nite cyclic subgroup with the relative number of ends

then splits along some virtually in?nite cyclic subgroup(except for some virtual surface groups).

Swarup and Scott proved the torsion free case.Bowditch’s proof is very different and uses the boundary.It also shows that the existence of such a splitting is a geometric property,i.e.,invariant under quasi-isometry.

The number of relative ends is given by taking a connected space on which acts freely and cocompactly as before and counting the ends of.One must exclude virtual surface groups in the above theorem,since they include triangle groups which do not split despite having cyclic subgroups with.

Rips and Sela have shown that the JSJ splittings are unique up to an appropriate equivalence relation in special cases.

18Geometric Group Theory

5.1Regular languages,?nite state automata,and automatic structures A ?nite state automaton over an alphabet is a ?nite directed,labeled graph equipped with a base vertex and a set of preferred vertices or “accept vertices.”The edges of are labeled by letters of ,where is an additional symbol that stands for “empty”.A ?nite state automaton de?nes a language .is the set of words labelling paths in starting from the base vertex of and ending at accept vertices of .For example,the language de?ned by the following ?nite state automaton is

.a

base vertex all vertices are accept vertices

A language is a regular language if it is the language of some ?nite state automaton.One also speaks of the language accepted by the ?nite state automaton.

We say the ?nite state automaton

is deterministic if no -edges occur and each vertex of has exactly one edge emanating from it for each element of .

A ?nite state automaton is a model of a simple computing device.This is best seen in the deterministic case.A deterministic ?nite state automaton can be written as a 5-tuple.,where is the alphabet,is the set of vertices,or states ,is the transition function given by the edges of ,is the base vertex

or start state and

is the set of preferred vertices or accept states .We imagine that performs computations on words to determine whether or not they lie in .Given the word ,the computation proceeds as follows:starts in the state and reads .This causes it to enter state .It proceeds in this way reading the letters of and changing states according to the formula .It concludes that if and only if .There are many characterizations known for regular languages.The following theo-rem,which gives some of them,is a worthwhile exercise.First a de?nition that we will return to later.We de?ne the cone of a word with respect to a language .It is the set Theorem/Exercise.The following are equivalent for a language .

i).is a regular language (i.e.,accepted by a ?nite state automaton);

ii).There are only ?nitely many different cones as runs through ;

iii).is accepted by some deterministic ?nite state automaton.(Hint:to show ii)iii)use the set of cones as the states of the ?nite state automaton.)Another standard way to characterize regular languages is in terms of closure prop-erties.We will not discuss this characterization,but it is a good exercise to prove the closure properties.

Geometric Group Theory19 Theorem.If are regular languages then so are:,,

,,and with.

In addition to using?nite state automaton to examine single words,we will need to use them to examine pairs.In order to do this,we use a technical trick:We form the padded product alphabet

where is a symbol not in.We then embed into by writing as

,if,

,if,and as

if.

When we speak of a subset of being the language of a?nite state automaton we mean it with respect to the padded product alphabet as above.A?nite state automaton for such a language is often called a(synchronous)two-tape?nite state automaton,since one imagines it being fed the two input words on two separate“input tapes.”If we extend the alphabet to include also pairs and,where denotes the empty word,one obtains what is called an asynchronous two-tape automaton,since the automaton can read a letter on only one tape at a time and thus read the two“tapes”asynchronously.

Another useful closure property of regular languages(again an exercise to prove)is: Proposition.If is the language of a(possibly asynchronous)two-tape automaton,then its projection onto the?rst factor is a regular language.

We now give the original automaton-theoretic de?nition of an automatic structure for.The characterization of the?rst theorem of 5.2gives a more useful working de?nition and is often used nowadays.

An automatic structure for consists of the following:

1)A?nite set together with a map.We write this map

.

3)A synchronous two-tape automaton so that

4)For each a synchronous two-tape automaton so that

Such a structure ful?ls our desire to build the Cayley graph“on the cheap".The language gives us names for the vertices of,and since it is regular,it is cheap to determine when we have such a name.cheaply determines when two names name the same vertex,and cheaply determines when two such names name an edge.All this can be made even cheaper by proving:

20Geometric Group Theory

Theorem.Any automatic structure for contains a sub-language which is an auto-matic structure which bijects to.

For a proof of this and other basics of automatic structures see[ECHLPT]or[BGSS].

5.2Fellow traveler property

There is a geometric condition which tells us when a regular language which surjects to the group is an automatic structure.We say that has the fellow traveler property if there exists such that for any pair of words so that and end at most an edge apart,for all.

Theorem.Suppose is a regular language that surjects to.Then is an automatic structure if and only if has the fellow traveler property.

Proof.We?rst show that an automatic structure has the fellow-traveler property.This uses the following simple but basic lemma about regular languages.

Lemma.Let be a regular language.Then there exists a bound such that if is an initial segment of an-word,that is,there exists a with,then there exists such a of length at most.

Proof.Let be the number of states in an automaton for.When we feed to the automaton,the fact that it can be extended to an-word means it ends at a state from which an accept state can be reached.This accept state can then clearly be reached in at most steps(in fact suf?ces).

Applying this lemma to the language accepted by the comparator automaton shows that a pair of words being compared are always at most steps away from being letter apart,so their values are at most apart,proving the fellow traveler property.

The fact that the fellow-traveler property implies the existence of comparator automata follows easily from the following proposition,and is left to the reader.

Proposition.Let be a group with?nite generating set.Then for any and any the languages

and-fellow-travel

are regular languages(actually,to be precise,we must use the corresponding padded languages,as described above).

Proof.Denote for each.Note that if we know and know the next letters and of and respectively then we can compute,since

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1 平面几何 一、常用定理（仅给出定理，证明请读者完成） 梅涅劳斯定理 设',','C B A 分别是ΔABC 的三边BC ，CA ，AB 或其延长线上的点，若',','C B A 三点共线，则 .1''''''=??B C AC A B CB C A BA 梅涅劳斯定理的逆定理 条件同上，若.1''''''=??B C AC A B CB C A BA 则',','C B A 三点共线。 塞瓦定理 设',','C B A 分别是ΔABC 的三边BC ，CA ，AB 或其延长线上的点，若',','CC BB AA 三线平行或共点，则.1''''''=??B C AC A B CB C A BA 塞瓦定理的逆定理 设',','C B A 分别是ΔABC 的三边BC ，CA ，AB 或其延长线上的点，若.1''''''=??B C AC A B CB C A BA 则',','CC BB AA 三线共点或互相平行。 角元形式的塞瓦定理 ',','C B A 分别是ΔABC 的三边BC ，CA ，AB 所在直线上的点，则',','CC BB AA 平行或共点的充要条件是.1'sin 'sin 'sin 'sin 'sin 'sin =∠∠?∠∠?∠∠BA B CBB CB C ACC AC A BAA 广义托勒密定理 设ABC D 为任意凸四边形，则AB ?CD+BC ?AD ≥AC ?BD ，当且仅当A ，B ，C ，D 四点共圆时取等号。 斯特瓦特定理 设P 为ΔABC 的边BC 上任意一点，P 不同于B ，C ，则有 AP 2=AB 2?BC PC +AC 2?BC BP -BP ?PC. 西姆松定理 过三角形外接圆上异于三角形顶点的任意一点作三边的垂线，则三垂足共线。 西姆松定理的逆定理 若一点在三角形三边所在直线上的射影共线，则该点在三角形的外接圆上。 九点圆定理 三角形三条高的垂足、三边的中点以及垂心与顶点的三条连线段的中点，这九点共圆。 蒙日定理 三条根轴交于一点或互相平行。（到两圆的幂（即切线长）相等的点构成集合为一条直线，这条直线称根轴） 欧拉定理 ΔABC 的外心O ，垂心H ，重心G 三点共线，且.2 1GH OG = 二、方法与例题 1．同一法。即不直接去证明，而是作出满足条件的图形或点，然后证明它与已知图形或点重合。 例1 在ΔABC 中，∠ABC=700，∠ACB=300，P ，Q 为ΔABC 内部两点，∠QBC=∠QCB=100，∠ PBQ=∠PCB=200，求证：A ，P ，Q 三点共线。 [证明] 设直线CP 交AQ 于P 1，直线BP 交AQ 于P 2，因为∠ACP=∠PCQ=100，所以 CQ AC QP AP =1 ，①在ΔABP ，ΔBPQ ，ΔABC 中由正弦定理有

高中数学立体几何讲义一

平面与空间直线 (Ⅰ)、平面的基本性质及其推论 图形 符号语言 文字语言（读法） 点A 在直线a 上。 点A 不在直线a 上。 点A 在平面α内。 点A 不在平面α内。 直线a 、b 交于A 点。 直线a 在平面α内。 直线a 与平面α无公共点。 直线a 与平面α交于点A 。 平面α、β相交于直线l 。 α（平面α外的直线）表示α或a A α=。 2、平面的基本性质 公理1： 如果一条直线的两点在一个平面内，那么这条直线上的所有点都在这个平面内 推理模式： A A B B ααα∈? ??∈? 。 如图示： 应用：是判定直线是否在平面内的依据，也是检验平面的方法。 公理2：如果两个平面有一个公共点,那么它们还有其他公共点,且所有这些公共点的集合是一条过这个公共点的直线。 推理模式：A l A αα ββ∈? ?=?∈? 且A l ∈且l 唯一如图示： 应用：①确定两相交平面的交线位置；②判定点在直线上。 例1．如图，在四边形ABCD 中，已知AB ∥CD ，直线AB ，BC ，AD ，DC 分别与平面 α相交于点E ，G ，H ，F ．求证：E ，F ，G ，H 四点必定共线． 解：∵AB ∥CD ， B A α D C B A

∴AB ，CD 确定一个平面β． 又∵AB α＝E ，AB ?β，∴E ∈α，E ∈β， 即E 为平面α与β的一个公共点． 同理可证F ，G ，H 均为平面α与β的公共点． ∵两个平面有公共点，它们有且只有一条通过公共点的公共直线， ∴E ，F ，G ，H 四点必定共线． 说明：在立体几何的问题中，证明若干点共线时，常运用公理2，即先证明这些点都是某二平面的公共点，而后得出这些点都在二平面的交线上的结论． 例2．如图，已知平面α，β，且α β＝l ．设梯形ABCD 中，AD ∥BC ，且AB ?α，CD ?β，求证：AB ，CD ，l 共点（相交于一点）． 证明 ∵梯形ABCD 中，AD ∥BC ， ∴AB ，CD 是梯形ABCD 的两条腰． ∴ AB ，CD 必定相交于一点， 设AB CD ＝M ． 又∵AB ?α，CD ?β，∴M ∈α，且M ∈β．∴M ∈α β． 又∵α β＝l ，∴M ∈l ， 即AB ，CD ，l 共点． 说明：证明多条直线共点时，一般要应用公理2，这与证明多点共线是一样的． 公理3： 经过不在同一条直线上的三点，有且只有一个平面。 推理模式：,, A B C 不共线?存在唯一的平面α，使得,,A B C α∈。 应用：①确定平面；②证明两个平面重合 。 例3．已知：a ，b ，c ，d 是不共点且两两相交的四条直线，求证：a ，b ，c ，d 共面． 证明 1o 若当四条直线中有三条相交于一点，不妨设a ，b ，c 相交于一点A ， 但A ?d ，如图1． ∴直线d 和A 确定一个平面α． 又设直线d 与a ，b ，c 分别相交于E ，F ，G ， 则A ，E ，F ，G ∈α． ∵A ，E ∈α，A ，E ∈a ，∴a ?α． 同理可证b ?α，c ?α． ∴a ，b ，c ，d 在同一平面α内． 2o 当四条直线中任何三条都不共点时，如图2． ∵这四条直线两两相交，则设相交直线a ，b 确定一个平面α． 设直线c 与a ，b 分别交于点H ，K ，则H ，K ∈α． 又 H ，K ∈c ，∴c ?α． 同理可证d ?α． ∴a ，b ，c ，d 四条直线在同一平面α内． α b a d c G F E A 图1 a b c d α H K 图2 α D C B A l 例2 β M

高中数学立体几何之面面平行的判定与性质讲义及练习

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F E " B A P C 3.如图，在四棱锥ABCD P -中，底面ABCD 是正方形， PA ⊥平面ABCD ， E 是PC 中点，F 为线段AC 上一点. （Ⅰ）求证：EF BD ⊥； （Ⅱ）试确定点F 在线段AC 上的位置，使EF PBD 4. 在四棱锥P ABCD 中， AB CD AB AD 4,22,2AB AD CD PA ABCD 4PA / （Ⅰ）设平面 PAB 平面 PCD m =，求证： CD m BD ⊥PAC Q PB QC PAC 33PQ PB 在如图所示的几何体中，四边形ABCD 为平行四边形，=90ABD ∠?， EB ⊥平面ABCD ， EF//AB ，2AB=，=1EF ，=13BC ，且M 是BD 的中点. （Ⅰ）求证：//EM 平面ADF ； （Ⅱ）在EB 上是否存在一点P ，使得CPD ∠最大 & 若存在，请求出CPD ∠的正切值；若不存在， 请说明理由. 6.如图，矩形ABCD 中，3AB =，4=BC ．E ，F 分别在线段BC 和AD 上，EF ∥AB ，将矩形ABEF 沿EF 折起．记折起后的矩形为MNEF ，且平面⊥MNEF 平面ECDF ． （Ⅰ）求证：NC ∥平面MFD ； （Ⅱ）若3EC =，求证：FC ND ⊥； （Ⅲ）求四面体NFEC 体积的最大值． ？ 7 如图1，在边长为3的正三角形ABC 中，E ，F ，P 分别为AB ，AC ，BC 上的点，且满足 P D C B A C A F E B M D

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