Infinite monoids as geometric objects

Robert Gray(joint work with Mark Kambites)

University of Essex, March 2014

Groups, monoids, and geometry

Gromov - “Infinite groups as geometric objects” InternationalCongress of Mathematicians address in Warsaw, 1984There are two main inter-related strands in geometric group theory

1. one seeks to understand groups by studying their actions on appropriatespaces, and

2. one seeks understanding from the intrinsic geometry of finitelygenerated groups endowed with word metrics.

How about monoids and semigroups?

1. To what extent can we gain information about finitely generatedmonoids by studying their actions on geometric objects?

2. How much algebraic information about finitely generated monoids isencoded in the geometry of their Cayley graphs?

Groups, monoids, and geometry

Gromov - “Infinite groups as geometric objects” InternationalCongress of Mathematicians address in Warsaw, 1984There are two main inter-related strands in geometric group theory

1. one seeks to understand groups by studying their actions on appropriatespaces, and

2. one seeks understanding from the intrinsic geometry of finitelygenerated groups endowed with word metrics.

How about monoids and semigroups?

1. To what extent can we gain information about finitely generatedmonoids by studying their actions on geometric objects?

2. How much algebraic information about finitely generated monoids isencoded in the geometry of their Cayley graphs?

General philosophy

Algebra Combinatorics

Groups Graphs

Monoids / Semigroups Digraphs

?? = directed metric spaces = semimetric spaces

Semimetric space = a set equipped with an asymmetric, partially-defineddistance function.

General philosophy

Algebra Combinatorics Geometry

Groups Graphs Metric spaces

Monoids / Semigroups Digraphs ??

?? = directed metric spaces = semimetric spaces

Semimetric space = a set equipped with an asymmetric, partially-defineddistance function.

General philosophy

Algebra Combinatorics Geometry

Groups Graphs Metric spaces

Monoids / Semigroups Digraphs ??

?? = directed metric spaces = semimetric spaces

Semimetric space = a set equipped with an asymmetric, partially-defineddistance function.

Cayley graphs and the notion of quasi-isometry

G - group, generated by a finite set A Ď G,Assume 1 R A, and A “ A´1.

This gives rise to a metric space pG, dAq with word metric dA.Points: GDistance: dApg, hq the minimum length of a word a1a2 ¨ ¨ ¨ ar P A˚ with theproperty that ga1a2 ¨ ¨ ¨ ar “ h.

The Cayley graph ΓpG,AqVertices: GEdges: g „ hô h “ ga for some a P A

The (?) Cayley graph of a group0-1 1 2 3-2-3

Z “ x˘t1uy

Z “ x˘t2, 3uy

ConclusionChanging the finite generating set can result in spaces that are not isometric.

Idea: These two spaces look the same when viewed from far enough away.This idea is formalised via the notion of quasi-isometry.

The (?) Cayley graph of a group0-1 1 2 3-2-3

Z “ x˘t1uy

Z “ x˘t2, 3uy

ConclusionChanging the finite generating set can result in spaces that are not isometric.

Idea: These two spaces look the same when viewed from far enough away.This idea is formalised via the notion of quasi-isometry.

Quasi-isometry for metric spaces

DefinitionLet pX, dXq and pY, dYq be two metric spaces. A map f : X Ñ Y is aquasi-isometric embedding if there exist constants λ ě 1 and C ě 0 suchthat

1λ

dXpa, bq ´ C ď dYpf paq, f pbqq ď λdXpa, bq ` C,

for all a, b P X.

pX, dXq and pY, dYq are quasi-isometric if in addition there is a constantD ě 0 such that every point in Y has a distance at most D from some point inthe image f pXq.

§ Quasi-isometry is an equivalence relation between metric spaces, whichignores finite details.

Quasi-isometry for metric spaces

DefinitionLet pX, dXq and pY, dYq be two metric spaces. A map f : X Ñ Y is aquasi-isometric embedding if there exist constants λ ě 1 and C ě 0 suchthat

1λ

dXpa, bq ´ C ď dYpf paq, f pbqq ď λdXpa, bq ` C,

for all a, b P X.

pX, dXq and pY, dYq are quasi-isometric if in addition there is a constantD ě 0 such that every point in Y has a distance at most D from some point inthe image f pXq.

§ Quasi-isometry is an equivalence relation between metric spaces, whichignores finite details.

Quasi-isometry for metric spaces

DefinitionLet pX, dXq and pY, dYq be two metric spaces. A map f : X Ñ Y is aquasi-isometric embedding if there exist constants λ ě 1 and C ě 0 suchthat

1λ

dXpa, bq ´ C ď dYpf paq, f pbqq ď λdXpa, bq ` C,

for all a, b P X.

pX, dXq and pY, dYq are quasi-isometric if in addition there is a constantD ě 0 such that every point in Y has a distance at most D from some point inthe image f pXq.

§ Quasi-isometry is an equivalence relation between metric spaces, whichignores finite details.

Quasi-isometry for finitely generated groups

PropositionLet A and B be two finite generating sets for the group G.Then the metric spaces pG, dAq and pG, dBq are quasi-isometric.

The quasi-isometry class of a groupGiven a finitely generated group G, the metric space pG, dAq is well definedup to quasi-isometry by the group G alone.

In particular, given two finitely generated groups G and H, one may askwhether they are quasi-isometric or not, without reference to any specificchoice of finite generating sets.

Examples of quasi-isometric groups

The quasi-isometry class of the trivial group G “ t1u is precisely the classof all finite groups.

Let G be a finitely generated group and H be a subgroup of finite index in G.Then H is finitely generated and quasi-isometric to G.

Let G be a finitely generated group and let N be a finite normal subgroup ofG. Then G{N is a finitely generated group quasi-isometric to G.

For k, l ě 2, the free groups Fk and Fl are quasi-isometric to each other.

Free abelian groups Zm and Zn are quasi-isometricô m “ n.

Tigers, Lions and Frogs

Tigers and lions look similar and,genetically, they have a lot oncommon.

Tigers and frogs on the other hand...

Quasi-isometric groups look similarand, algebraically, they have a lot incommon.

Tigers, Lions and Frogs

Tigers and lions look similar and,genetically, they have a lot oncommon.

Tigers and frogs on the other hand...

Quasi-isometric groups look similarand, algebraically, they have a lot incommon.

Quasi-isometry invariants

DefinitionA property (P) of finitely generated groups is said to be a quasi-isometryinvariant if, whenever G1 and G2 are quasi-isometric finitely generatedgroups,

G1 has property (P)ðñ G2 has property (P).

Examples of quasi-isometry invariants(i) Finite (ii) Infinite virtually cyclic (iii) Finitely presented (iv) Virtuallyabelian (v) Virtually nilpotent (vi) Virtually free (vii) Amenable (viii)Hyperbolic (ix) Accessible (x) Type of growth (xi) Finitely presented withsolvable word problem (xii) Satisfying the homological finiteness conditionFn or the condition FPn (xiii) Number of ends.

General philosophy

Algebra Combinatorics Geometry

Groups Graphs Metric spaces

Monoids / Semigroups Digraphs ??

?? = directed metric spaces = semimetric spaces

Semimetric space = a set equipped with an asymmetric, partially-defineddistance function.

Cayley graphs of semigroups and monoids1

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((hh b ))

ii b2 **ii b3 **

ii b4 **ii b5

c

��

))hh cb

**ii cb2 **

jj cb3 **jj cb4 **

jj cb5

c2

��

**ii c2b

++jj c2b2 ++

kk c2b3 ++kk c2b4 ++

kk c2b5

c3

��

**ii c3b

++jj c3b2 ++

kk c3b3 ++kk c3b4 ++

kk c3b5

c4

��

**ii c4b

++jj c4b2 ++

kk c4b3 ++kk c4b4 ++

kk c4b5

c5 **ii c5b

++jj c5b2 ++

kk c5b3 ++kk c5b4 ++

kk c5b5

The bicyclic monoid B “ xb, c | bc “ 1y

Semimetric spaces

Definition (Semimetric space)A semimetric space is a pair pX, dq where X is a set, and

d : X ˆ X Ñ R8 “ Rě0 Y t8u

is a function satisfying:(i) dpx, yq “ 0 if and only if x “ y

(ii) dpx, zq ď dpx, yq ` dpy, zq

for all x, y, z P X.

Here R8 “ Rě0 Y t8u with the obvious order, and we set

8` x “ x`8 “ y8 “ 8y “ 8

for all x P R8 and y P R8zt0u.

Monoids as semimetric spaces

M - monoid generated by a finite set A.

This gives rise to a semimetric space pM, dAq with word semimetric dA.Points: MDirected distance: dApx, yq the minimum length of a word a1a2 ¨ ¨ ¨ ar P A˚

with the property that xa1a2 ¨ ¨ ¨ ar “ y, or8 if there is no such word.

The (right) Cayley graph ΓpM,AqVertices: MDirected edges: x Ñ y iff y “ xa for some a P A

Quasi-isometry for semimetric spaces

DefinitionLet pX, dXq and pY, dYq be two semimetric spaces. A map f : X Ñ Y is aquasi-isometric embedding if there exist constants λ ě 1 and C ě 0 suchthat

1λ

dXpa, bq ´ C ď dYpf paq, f pbqq ď λdXpa, bq ` C,

for all a, b P X.

The semimetric spaces pX, dXq and pY, dYq are quasi-isometric if in additionthere is a constant D ě 0 such that for every y P Y there exists a z P f pXqsuch that

dYpy, zq ď D and dYpz, yq ď D.

§ Quasi-isometry is an equivalence relation between semimetric spaces.

Quasi-isometry for semimetric spaces

DefinitionLet pX, dXq and pY, dYq be two semimetric spaces. A map f : X Ñ Y is aquasi-isometric embedding if there exist constants λ ě 1 and C ě 0 suchthat

1λ

dXpa, bq ´ C ď dYpf paq, f pbqq ď λdXpa, bq ` C,

for all a, b P X.

The semimetric spaces pX, dXq and pY, dYq are quasi-isometric if in additionthere is a constant D ě 0 such that for every y P Y there exists a z P f pXqsuch that

dYpy, zq ď D and dYpz, yq ď D.

§ Quasi-isometry is an equivalence relation between semimetric spaces.

Quasi-isometry for finitely generated monoids

PropositionLet A and B be two finite generating sets for a monoid M.Then the semimetric space pM, dAq is quasi-isometric to the semimetricspace pM, dBq.

The quasi-isometry class of a monoidThe semimetric space pM, dAq is well defined up to quasi-isometry by thefinitely generated monoid M alone.

In particular, given two finitely generated monoids M and N, one may askwhether they are quasi-isometric or not, without reference to any specificchoice of finite generating sets.

Quasi-isometric monoids

Quasi-isomtries between monoids preserve right ideal structure.

Two finite monoids M and N are quasi-isometric if and only if the partiallyordered sets or R-classes M{R and N{R are isomorphic.

If M is a finitely generated monoid, and η is a congruence on M, then M andM{η are quasi-isometric if there is a bound on the diameter of the η-classesof M.

Finitely generated free monoids are quasi-isometric if and only if they havethe same rank.

Finitely generated free commutative monoids are quasi-isometric if and onlyif they have the same rank.

Quasi-isometry invariants of monoids

TheoremThe following properties are all quasi-isometry invariants of finitelygenerated monoids:

§ Finiteness;§ Number of right ideals;§ Being a group (for monoids);§ Being right simple (for semigroups);§ Type of growth;§ Number of ends (in the sense of Jackson and Kilibarda (2009)).

Finite presentability and the word problem

Q1 Is finite presentability a quasi-isometry invariant of finitely generatedmonoids?

Q2 Is being finitely presented with solvable word problem a quasi-isometryinvariant of finitely generated monoids?

One should not expect yes to be the answer for monoids in general. Indeed:§ Jenni Awang (St Andrews) recently found a counterexample to (Q1);§ (Q2) is still open, but we have constructed an example to show that

having solvable word problem is not a quasi-isometry invariant offinitely generated monoids.

Idea: Investigate (Q1) and (Q2) for classes of monoids that lie betweenmonoids and groups, such as:

1. Left cancellative monoids2. Monoids with finitely many left and right ideals

Obviously any group satisfies both (1) and (2).

Finite presentability and the word problem

Q1 Is finite presentability a quasi-isometry invariant of finitely generatedmonoids?

Q2 Is being finitely presented with solvable word problem a quasi-isometryinvariant of finitely generated monoids?

One should not expect yes to be the answer for monoids in general. Indeed:§ Jenni Awang (St Andrews) recently found a counterexample to (Q1);§ (Q2) is still open, but we have constructed an example to show that

having solvable word problem is not a quasi-isometry invariant offinitely generated monoids.

Idea: Investigate (Q1) and (Q2) for classes of monoids that lie betweenmonoids and groups, such as:

1. Left cancellative monoids2. Monoids with finitely many left and right ideals

Obviously any group satisfies both (1) and (2).

Left cancellative semigroups and monoids

Left cancellativity: ab “ ac ñ b “ c.Right cancellativity, and cancellativity are defined analogously.

Interesting classes of cancellative monoids

§ Divisibility monoids (Droste & Kuske (2001));§ Garside monoids; includes, spherical Artin monoids, Braid monoids of

complex reflection groups etc. (Dehornoy & Paris (1999)).

One-relator monoids§ Adyan and Oganesyan (1987): Decidability of the word problem for

one relator monoids is reducible to the left cancellative case.§ Motivates the development of new methods for approaching the word

problem for finitely presented left cancellative monoids.

Directed 2-complexes (Guba & Sapir (2006))

DefinitionA directed 2-complex is a digraph Γ, together with a set F of faces.

2-paths = paths between paths

K - a directed 2-complex, with underlying digraph Γ, and set of faces FThe 1-paths in K are the paths in Γ.

Definition (2-path)An atomic 2-path δ is a triple pp, f , qq where p, q are 1-paths, f P F and:

p qf

rf s

tf u

A 2-path in K is then a sequence δ “ δ1δ2 . . . δn of composable atomic2-paths.

Two paths p, q in K are homotopic if there is a 2-path between them.

2-path example

f1

f2 f3

f4

a

a

a b

b

a

b a

a bb

b a

a

Γ - right Cayley graph of the monoid xa, b | ab “ bay.

K4pΓq - directed 2-complex with underlying digraph Γ and face set F givenby adding a face for every pair p ‖ q of parallel paths with |p| ` |q| ď 4.

Diagram illustrates a 2-path δ of length 4 in K4pΓq.

2-path example

f1

f2 f3

f4

a

a

a b

b

a

b a

a bb

b a

a

Γ - right Cayley graph of the monoid xa, b | ab “ bay.

K4pΓq - directed 2-complex with underlying digraph Γ and face set F givenby adding a face for every pair p ‖ q of parallel paths with |p| ` |q| ď 4.

Diagram illustrates a 2-path δ of length 4 in K4pΓq.

2-path example

f1

f2 f3

f4

a

a

a b

b

a

b a

a bb

b a

a

Γ - right Cayley graph of the monoid xa, b | ab “ bay.

K4pΓq - directed 2-complex with underlying digraph Γ and face set F givenby adding a face for every pair p ‖ q of parallel paths with |p| ` |q| ď 4.

Diagram illustrates a 2-path δ of length 4 in K4pΓq.

2-path example

f1

f2 f3

f4

a

a

a b

b

a

b a

a bb

b a

a

Γ - right Cayley graph of the monoid xa, b | ab “ bay.

K4pΓq - directed 2-complex with underlying digraph Γ and face set F givenby adding a face for every pair p ‖ q of parallel paths with |p| ` |q| ď 4.

Diagram illustrates a 2-path δ of length 4 in K4pΓq.

2-path example

f1

f2 f3

f4

a

a

a b

b

a

b a

a bb

b a

a

Γ - right Cayley graph of the monoid xa, b | ab “ bay.

K4pΓq - directed 2-complex with underlying digraph Γ and face set F givenby adding a face for every pair p ‖ q of parallel paths with |p| ` |q| ď 4.

Diagram illustrates a 2-path δ of length 4 in K4pΓq.

2-path example

f1

f2 f3

f4

a

a

a b

b

a

b a

a bb

b a

a

Γ - right Cayley graph of the monoid xa, b | ab “ bay.

K4pΓq - directed 2-complex with underlying digraph Γ and face set F givenby adding a face for every pair p ‖ q of parallel paths with |p| ` |q| ď 4.

Diagram illustrates a 2-path δ of length 4 in K4pΓq.

Directed homotopy and simple connectedness

Directed homotopyA directed 2-complex K is directed simply connected if for every pair p ‖ qof parallel paths, p and q are homotopic in K.

Quasi-simple connectednessΓ - digraph, n P NKnpΓq “ directed 2-complex with underlying digraph Γ and face set

F “ tpp, qq | p and q are parallel paths in Γ with |p| ` |q| ď nu.

We say the digraph Γ is quasi-simply-connected if KnpΓq is directed simplyconnected for some n.

Note: KnpΓq is the natural directed analogue of the Rips complex fromgeometric group theory.

Directed homotopy and simple connectedness

Directed homotopyA directed 2-complex K is directed simply connected if for every pair p ‖ qof parallel paths, p and q are homotopic in K.

Quasi-simple connectednessΓ - digraph, n P NKnpΓq “ directed 2-complex with underlying digraph Γ and face set

F “ tpp, qq | p and q are parallel paths in Γ with |p| ` |q| ď nu.

We say the digraph Γ is quasi-simply-connected if KnpΓq is directed simplyconnected for some n.

Note: KnpΓq is the natural directed analogue of the Rips complex fromgeometric group theory.

Finite presentability and the word problem

TheoremLet S be a left cancellative monoid generated by a finite set A. Then:

§ S is finitely presentedô ΓpS,Aq is quasi-simply-connected.

PropositionThe property of being quasi-simply-connected is a quasi-isometry invariantof directed graphs.

TheoremLet M and N be left cancellative, finitely generated monoids which arequasi-isometric. Then M is finitely presentableô N is finitely presentable.

By defining and studying Dehn functions of directed 2-complexes and theirbehaviour under quasi-isometry, one can show:

TheoremLet M and N be left cancellative, finitely presentable monoids which arequasi-isometric. Then M has solvable word problem if and only if N hassolvable word problem.

Finite presentability and the word problem

TheoremLet S be a left cancellative monoid generated by a finite set A. Then:

§ S is finitely presentedô ΓpS,Aq is quasi-simply-connected.

PropositionThe property of being quasi-simply-connected is a quasi-isometry invariantof directed graphs.

TheoremLet M and N be left cancellative, finitely generated monoids which arequasi-isometric. Then M is finitely presentableô N is finitely presentable.

By defining and studying Dehn functions of directed 2-complexes and theirbehaviour under quasi-isometry, one can show:

TheoremLet M and N be left cancellative, finitely presentable monoids which arequasi-isometric. Then M has solvable word problem if and only if N hassolvable word problem.

Monoids with finitely many left and right ideals

We established a Švarc–Milnor Lemma for groups acting on geodesicsemimetric spaces. Applying this result to Schützenberger groups acting onSchützenberger graphs leads to the following.

TheoremLet M be a finitely generated monoid with finitely many left and right ideals.Then M is finitely presented if and only if all right Schützenberger graphs ofM are quasi-simply-connected.

TheoremFor finitely generated monoids with finitely many left and right ideals, finitepresentability is a quasi-isometry invariant.

GpHq ý R

H

Future directions

§ Are there other natural classes of monoids for which the properties ofbeing

(a) finitely presented;(b) finitely presented with solvable word problem;

are quasi-isometry invariants?§ Investigate other properties from the point of view of quasi-isometry

(e.g. Amenable semigroups (Day (1957)) / Følner conditions indigraphs and semimetric spaces).

§ We have restricted our attention to the geometry of right Cayley graphsonly. What if one considers the geometry of right and left Cayleygraphs simultaneously?