On abstract commensurators of surface groups
OON ABSTRACT COMMENSURATORS OF SURFACE GROUPS
KHALID BOU-RABEE AND DANIEL STUDENMUND
Abstract.
Let Γ be the fundamental group of a surface of finite type andComm(Γ) be its abstract commensurator. Then Comm(Γ) contains the solv-able Baumslag–Solitar groups h a, b : aba − = b n i for any n >
1. Moreover,the Baumslag–Solitar group h a, b : ab a − = b i has an image in Comm(Γ)that is not residually finite. Our proofs are computer-assisted.Our results also illustrate that finitely-generated subgroups of Comm(Γ)are concrete objects amenable to computational methods. For example, wegive a proof that h a, b : ab a − = b i is not residually finite without the use ofnormal forms of HNN extensions. Introduction
Let G be a group. The abstract commensurator of G , denoted Comm( G ), is theset of equivalence classes of isomorphisms φ : H → H for finite-index subgroups H , H ≤ G , where two isomorphisms are equivalent if they are both defined andequal on a common finite-index subgroup of G . The set Comm( G ) is a group undercomposition. Elements of Comm( G ) are called commensurators of G .The results of this paper serve two purposes. First, they shed light on the struc-ture of Comm(Γ) when Γ is the fundamental group of a surface of finite type.These abstract commensurator groups are not well-understood, and in particu-lar are known to be neither finitely generated [BB10] nor linear over C [Stu15,Proposition 7.6]. Second, the methods of proof suggest that maps into abstractcommensurators of fundamental groups of surfaces of finite type provide a new sortof “representation theory” of groups that have no faithful representations throughmatrices. This view has immediate utility: we give a new and concrete proof of aclassical result of Baumslag–Solitar concerning their one-relator groups [BS62].1.1. The main result.
The
Baumslag–Solitar groups are given by the presentationBS( m, n ) := (cid:10) a, b : ab m a − = b n (cid:11) . By surface group , we mean the fundamental group of a surface with finite genus, andfinitely many punctures and boundary components. Any surface group is finitelygenerated. Theorem 1.
Let Γ be a surface group, and let n > . Then the group Comm(Γ) contains
BS(1 , n ) . Moreover, Comm(Γ) contains a non-residually finite image of
BS(2 , . Date : March 8, 2019.2000
Mathematics Subject Classification.
Primary 20E26, 20B07; Secondary 20K10.
Key words and phrases. commensurators, algebraic groups, residually finite groups.DS supported in part by NSF Grant In the literature this is also known as the virtual automorphism group . a r X i v : . [ m a t h . G R ] J u l KHALID BOU-RABEE AND DANIEL STUDENMUND
Our proof of Theorem 1 appears in §2 with supplementary code and theoryappearing in the appendix. Since Comm( G ) is an invariant of the commensurabilityclass of G , Theorem 1 is a statement about the abstract commensurators of twogroups: the nonabelian free group of rank two, and the fundamental group of acompact surface of genus two.We prove the nonabelian free group case first, and then build upon it to handlethe compact case. The first part of each case uses some of the elementary theory ofsurface topology. The second part of each case requires some computer-assistance,using the GAP System for Computational Discrete Algebra [GAP17].The length of our proof belies the difficulty of the proof of the cocompact case,which relies on a long and delicate computation supplied in Appendix C. The appen-dix is a substantial part of the theory in this paper: the setup, hand computations,and computer computations find an explicit representation of a Baumslag–Solitargroup in Comm(Γ).1.2. On local residual finiteness.
A group is (locally) residually finite if every(finitely generated) subgroup is residually finite. Linear groups (e.g., fundamentalgroups of surfaces, GL n ( C ), or GL n ( F p [ t ])) and mapping class groups of surfaces(e.g., braid groups) are important classes of locally residually finite groups [Mal56,Gro75].Our main theorem determines precisely when the abstract commensurator of alattice in a semisimple Lie group is locally residually finite. See §3 for the proof. Corollary 2.
Let Γ be an irreducible lattice in a semisimple Lie group G withoutcompact factors. Then Comm(Γ) is locally residually finite if and only if G is notlocally isomorphic to PSL ( R ) . In particular, the abstract commensurator of anylattice in PSL n ( R ) is locally residually finite if and only if n = 2 . The fact that lattices in PSL ( R ) have abstract commensurators which are notlocally residually finite is surprising for two reasons. First, Baumslag showed thatfor any finitely-generated residually finite G , the group Aut( G ) (which Comm( G )can be thought to generalize) is residually finite [Bau63]. Second, Odden showedthat the abstract commensurator of the fundamental group of a closed surface isa mapping class group of an inverse limit of surfaces (the “universal hyperbolicsolenoid”) [Odd05].1.3. On residual finiteness of
BS(2 , . Our proof of Theorem 1 explicitly identi-fies an element γ ∈ BS(2 ,
3) in the kernel of a surjective homomorphism BS(2 , → BS(2 , γ has nontrivial image under a homomorphismBS(2 , → Comm(Γ). Thus, our methods prove that BS(2 ,
3) is not residuallyfinite without using the normal forms ensured by the Britton’s Lemma [Bri63].Bypassing this step makes the original proof that BS(2 ,
3) is not residually finite,due to Baumslag–Solitar [BS62], elementary and concrete. We emphasize that ourproof relies solely on the Comm( F ) case, and hence does not require computerverification. See §4 for the proof. Theorem 3 ([BS62]) . The group BS (2 , is not residually finite. On two groups.
Let F be the nonabelian free group of rank two. Let Γ bethe fundamental group of a compact surface of genus two. The groups Comm( F )and Comm(Γ ) share a number of properties: Bartholdi–Bogopolski showed thatneither group is finitely generated [BB10]. Moreover, finitely-generated subgroups N ABSTRACT COMMENSURATORS OF SURFACE GROUPS 3 of them have solvable word problem (see §1.5). The proof of Theorem 1 shows thatthey both contain many infinite images of Baumslag–Solitar groups.In spite of these similarities, in §5 we show that Comm( F ) contains more finitesubgroups than Comm(Γ ) to prove that they are not isomorphic. Proposition 4.
The groups
Comm( F ) and Comm(Γ ) are not isomorphic. On the word problem.
Although the groups Comm(Γ) are not well-understoodwhen Γ is a surface group, their finitely-generated subgroups may be understoodconcretely, as in the computations that appear in Appendix B. In particular, theword problem is solvable. A more general version of the following is proved in §6.
Proposition 5.
Finitely-generated subgroups of abstract commensurators of sur-face groups have solvable word problem.
Theorem 1 provides an example of a non-residually finite group G that faithfullyembeds in Comm( F ). Such G cannot be completely understood through its lin-ear representations, while words in its image in Comm( F ) can be evaluated bya computer. This shows that representations into Comm( F ) may be useful forunderstanding abstract groups without faithful linear representations. See §6 forfurther discussion and questions.1.6. Historical remarks.
Questions about abstract commensurators of surfacegroups have been asked for at least 30 years [Man87]. More generally, abstract com-mensurators detect arithmeticity of lattices in higher-rank semisimple Lie groups(see discussion after Corollary 2 in §1.2), give “generalized Hecke operators” fromnumber theory (see [Sha00, page 10]), play a role in extending linear group methodsto nonlinear settings (e.g., with mapping class groups [FH07, §1]), and are used togive more algebraic descriptions of existing infinite finitely presented simple groups(e.g., [R¨02]).Abstract commensurators can also be understood in the context of coarse ge-ometry. The quasi-isometry group
QI(Γ) of a group Γ is the group of equivalencesclasses of self-quasi-isometries f : Γ → Γ, up to coarse equivalence. There is anatural map Comm(Γ) → QI(Γ) which is injective when Γ is finitely generated[FM02, Theorem 7.4]. This shows, for example, that the abstract commensuratorof a word hyperbolic group acts faithfully on its Gromov boundary.That the abstract commensurator of a free group of rank two is not locally resid-ually finite is a well-known folklore result. The folklore proof relies on the existenceof a simple, finitely presented group that is isomorphic to an amalgamated productof two finitely-generated free groups over a common finite-index subgroup. Thefirst groups of this kind were constructed by Burger–Mozes [BM97]. It is unknownwhether any of the Burger–Mozes groups embed inside the abstract commensuratorof a closed oriented surface group of genus two.
Acknowledgements.
We are grateful to Benson Farb and Andrew Putman forconversations and support. We are especially grateful to Yves Cornulier and BensonFarb for giving us many useful comments that substantially improved the paper.2.
Proof of Theorem 1
Let F be the free group of rank two. Let Σ g be the compact oriented surfaceof genus g . Let Γ be a surface group. As stated in §1.1 and §1.4, the group KHALID BOU-RABEE AND DANIEL STUDENMUND
Comm(Γ) is isomorphic to either Comm( F ) or Comm( π (Σ , ∗ )). We handle eachcase separately, as they are different enough to warrant different proofs (the firstserving as an outline of the second).It will be important to us in each case that Γ satisfies the unique root property :if elements x, y ∈ Γ satisfy x n = y n for some n = 0, then x = y [BB10, Lemma2.2], [BH99, pages 462–463]. A consequence of the unique root property is thattwo isomorphisms f : A → B and g : C → D between finite-index subgroupsof Γ represent the same element of Comm(Γ) if and only if f | A ∩ C = g | A ∩ C . Inparticular, to check that f represents a nontrivial elements of Comm(Γ), it sufficesto find an element γ ∈ Γ such that f ( γ ) = γ .2.1. The
Comm( F ) case. We first describe a method for constructing images ofBS( n, m ) in Comm( F ). Set F = h A, B i . Let π : F → Z /m × Z /n be the mapgiven by A (1 ,
0) and B (0 , π : F → Z /m × Z /n be the map givenby A (0 ,
1) and B (1 , = ker( π ) and ∆ = ker( π ).Let φ be the commensurator with representative f : F → F given by f ( γ ) = AγA − . Let ψ be a commensurator with representative g : ∆ → ∆ , where g ( A m ) = A n ; such an isomorphism g exists because ∆ and ∆ are free groups ofthe same rank in which A m and A n , respectively, are elements in free generatingsets. Then the commensurator ψ ◦ φ m ◦ ψ − has a representative h = g ◦ f m ◦ g − such that for every γ ∈ ∆ , h ( γ ) = g ◦ f m ◦ g − ( γ ) = g ( A m g − ( γ ) A − m ) = A n ( g ◦ g − ( γ )) A − n = A n γA − n , and hence ψ ◦ φ m ◦ ψ − = φ n . Thus, the assignment a ψ and b φ defines ahomomorphism Φ : BS( m, n ) → Comm( F ).Next, we show that the map BS( m, n ) → h ψ, φ i is an isomorphism of groupswhen m = 1. Let z be in the kernel of this map. Then z = a s γ , where γ is inthe normal closure of b . Since hh b ii = h a t ba − t : t ≤ i , we have that z is conjugateto an element of the form a s b t for some integers s and t . It suffices to show that ψ s ◦ φ t is only trivial in Comm( F ) if s = t = 0.If s > ψ s has a representative that maps A to A n s . By the uniqueroot property of F , it is impossible for an automorphism of F to send A to A n s . Similarly, if s < ψ s has a representative that maps A n − s to A , whichcannot be extended to an automorphism of F . If ψ s ◦ φ t is trivial, then ψ s has arepresentative that is an automorphism of F , and so s = 0. It is clear that φ t istrivial only if t = 0 because f t ( B ‘ ) = A t B ‘ A − t for all ‘ . This complete the proofof the m = 1 case.To finish, we need to show that when m = 2 and n = 3, the image of BS( m, n )as defined above is not residually finite. To do this, we need to show that element γ := b − aba − b − aba − b − has nontrivial image. Indeed, it is in the kernel of the surjective map BS(2 , → BS(2 ,
3) given by a a and b b , and thus is in the residual finiteness kernel ofBS(2 , γ ) on the word BAB − A − ∈ F and checking that the result is not equal to BAB − A − . This isstraightforward, but tedious, to compute by hand. To perform this calculation, weused computer assistance. See the code in Appendix B and explanations there toend the proof of the Comm( F ) case. N ABSTRACT COMMENSURATORS OF SURFACE GROUPS 5
Figure 1.
Cover of Σ for §2.22.2. The
Comm( π (Σ , ∗ )) case. As in §2.1, we begin by describing how to obtainimages of BS( n, m ) inside Comm( π (Σ , ∗ )). Let [ X, Y ] :=
XY X − Y − and letΓ := π (Σ , ∗ ) = h A, B, C, D : [
A, B ][ C, D ] i . Set π : Γ → Z /m × Z /n be a map satisfying A (1 , π the curve corresponding to A m lifts to a non-separatingsimple closed curve. See Figure 1 for the cover corresponding to an example ofsuch a map. Similarly, set π : Γ → Z /m × Z /n be a map satisfying A (0 , π the curve corresponding to A n lifts to anon-separating simple closed curve. Set ∆ = ker( π ) and ∆ = ker( π ).Let φ be the commensurator with representative f : Γ → Γ given by f ( γ ) = AγA − . To define the commensurator ψ we need some additional setup:Let p : S → Σ and p : S → Σ be the covers corresponding to ∆ and ∆ respectively. Then A m and A n lift to non-separating simple-closed curves in S and S by construction. Any oriented surface group is uniquely determined by its Eulercharacteristic. Both S and S cover Σ with degree mn and hence have Eulercharacteristic − mn by the Riemann–Hurwitz formula. It follows that S and S are homeomorphic. Moreover, by the Change of Coordinates Principle [FM12, p.37] there is a homeomorphism S → S inducing an isomorphism g : ∆ → ∆ ,where g ( A m ) = A n . Let ψ be the commensurator with representative g . Then thecommensurator ψ ◦ φ n ◦ ψ − has a representative h = g ◦ f n ◦ g − such that forevery γ ∈ ∆ , h ( γ ) = g ◦ f m ◦ g − ( γ ) = g ( A m g − ( γ ) A − m ) = A n ( g ◦ g − ( γ )) A − n = A n γA − n , and hence ψ ◦ φ m ◦ ψ − = φ n . Thus, the assignment a ψ and b φ defines ahomomorphism Φ : BS( m, n ) → Comm(Γ ).Since Γ has the unique root property, the argument for showing that the mapBS( m, n ) → h ψ, φ i is an isomorphism when m = 1 in §2.1 applies here verbatim.Hence, it only remains to show that when m = 2 and n = 3 the element γ := b − aba − b − aba − b − See Appendix C for an explicit construction of such a map in the case ( m, n ) = (2 , KHALID BOU-RABEE AND DANIEL STUDENMUND has nontrivial image, since this is in the residual finiteness kernel of BS( m, n ). Therest of the proof, as before, is computer-assisted. The computer computations aremore difficult to implement here as surface groups do not have as much flexibilityas free groups. See Appendix C and explanations there to end the proof. (cid:3) Proof of Corollary 2
By Theorem 1 it suffices to show that the abstract commensurator of any irre-ducible lattice Λ in a semisimple Lie group G without compact factors and not lo-cally isomorphic to PSL ( R ) is locally residually finite. Indeed, for any such Λ ≤ G ,strong rigidity results of Mostow–Prasad–Margulis [Mar91] show that every com-mensurator of Λ extends to an automorphism of G . By the Borel density theorem[Bor60], the induced map Comm(Λ) → Aut( G ) is faithful. This shows Comm(Λ)is linear and thus locally residually finite by Malćev’s Theorem [Mal71]. (cid:3) Proof of Theorem 3
In §2.1 there is a concrete and elementary proof that the element γ := b − aba − b − aba − b − is not the identity in BS (2 , ρ : BS(2 , → BS(2 ,
3) given by a a and b b . It is a standard argumentthat this shows BS(2 ,
3) is not residually finite. We include it here for completeness:If BS(2 ,
3) were residually finite, then there would exist a finite group Q anda homomorphism f : BS(2 , → Q such that f ( γ ) = 1. For each k there is anelement γ k such that ρ k ( γ k ) = γ . Then γ k is not in the kernel of f ◦ ρ k , but lies inthe kernel of f ◦ ρ n for all n ≥ k , and hence the maps f ◦ ρ k : BS(2 , → Q are alldistinct for positive k . This is impossible since there are only finitely many mapsfrom a finitely-generated group to a fixed finite group. (cid:3) Proof of Proposition 4
We will show that the group Comm( F ) contains an infinite collection of groupsnot in Comm(Γ ). To start, we record that Comm( F ) contains every finite group.Every finitely-generated free group F n for n ≥ F , andbecause F has the unique root property, the natural maps Aut( F n ) → Comm( F )are injective. Every finite group lies in some symmetric group, hence in Aut( F n )for some n .However, Comm(Γ ) only contains finite groups that have a cyclic subgroup ofindex at most 2. Indeed, the Gromov boundary of Γ is S , and so Comm(Γ ) actsfaithfully on S as described in §1.6. (This action was originally observed by Odden[Odd05, Proposition 4.5].) Since any square and any commutator in Homeo( S ) isorientation preserving, we see that [Homeo( S ) : Homeo + ( S )] = 2. Moreover, allfinite subgroups of Homeo + ( S ) are cyclic [BS18, Lemma 3.1], it follows that everyfinite subgroup of Comm(Γ ) contains a cyclic subgroup of index at most 2. (cid:3) N ABSTRACT COMMENSURATORS OF SURFACE GROUPS 7 A parting proof with questions
Proposition 6.
Let Γ be a finitely presented group with solvable word problem andthe unique root property. Then any finitely generated subgroup of Comm(Γ) hassolvable word problem.Proof.
Fix a finite set S of elements in Comm(Γ). We outline an algorithm thatinputs a word in S and outputs “True” if the word is the identity, and “False”otherwise. We take as constants in our algorithm representatives for each elementin S , along with indices and finite generating sets for each of the domains andcodomains of elements in S (such generating sets exist by the Reidemeister-SchreierMethod). Given the above constants, the inverse of each element in S can becomputed, so without loss of generality we assume that S is symmetric.We start with the simplest case, when the word is of length one. Let g : A → B be the representative of the commensurator of Γ corresponding to the letter,where A, B are of finite index inside Γ and have explicitly written generating sets.The isomorphism g is fully determined by its values on the fixed finite generatingset for A . Since the word problem in Γ is solvable, determining whether g fixeseach generator of A is solvable, and hence determining whether g is the identityis solvable. It follows, by our assumptions on Γ, that determining whether thecommensurator induced by g is the identity is solvable.For a word of length two we have two maps g : A → B and h : C → D , where A, B, C, D are all of finite index inside Γ and have explicitly written generating setsand indices. Define g ? h = g | A ∩ D ◦ h | h − ( A ∩ D ) which is a map h − ( A ∩ D ) → g ( A ∩ D ) representing the composition of commen-surators [ g ] ◦ [ h ] ∈ Comm(Γ). Because Γ has the unique root property, [ g ] ◦ [ h ] istrivial in Comm(Γ) if and only if g ? h is the identity function. Since Γ is finitelypresented, there is an algorithm for finding a finite quotient of Γ whose kernel iscontained in A ∩ D . Thus, there is an algorithm for computing a finite generatingset for A ∩ D , and subsequently h − ( A ∩ D ). As the homomorphism g ? h is deter-mined by its values on a generating set of h − ( A ∩ D ), determining whether g ? h is the identity is solvable because Γ has solvable word problem.Repeating this process inductively for arbitrary compositions h ? h ? h ? · · · ? h k for letters h i , allows one to handle word of arbitrarily length, giving the desiredalgorithm. (cid:3) G that embeds inside Comm( F ) butnot inside Comm(Γ )? Vice versa?(3) Let Γ be a surface group. Do finitely generated subgroups of Comm(Γ)satisfy a Tits’ alternative? KHALID BOU-RABEE AND DANIEL STUDENMUND
Note that a partial answer to Question 3 is given in the compact surface group caseby [Mar00], using the fact that Comm(Γ) embeds in Homeo( S ) [Odd05]. Appendix A. The GAP System
In the appendices that follow we use the GAP System to complete our proofs.Borrowing words from the creators [GAP17]: “GAP is a system for computationaldiscrete algebra, with particular emphasis on Computational Group Theory. GAPprovides a programming language, a library of thousands of functions implementingalgebraic algorithms written in the GAP language as well as large data libraries ofalgebraic objects.” GAP is especially well-suited for our needs, with key functionsthat we will use to explicitly evaluate commensurators. Here is a short glossary ofsome of the key functions used in our code:
GroupHomomorphismByImages ( domain, codomain, list1, list2 ). Inputs twogroups “domain” and “codomain” with lists “list1” and “list2”. When GAP runsthis command it first verifies that a well-defined homomorphism from “domain” to“codomain” sending “list1” to “list2” exists, returning “fail” otherwise. If the ho-momorphism requested is well-defined, this returns a homomorphism from domainto codomain where elements of “list1” consisting of generators of domain are sentto corresponding elements in “list2”.
Image (map, elem). Inputs a homomorphism “map” and an element from the do-main “elem”. Returns the image of element “elem” under an application of thehomomorphism “map”.
InverseGeneralMapping (map). Inputs a homomomorphism “map”. This func-tion returns an inverse of an isomorphism of groups where the domain and codomainare not equal.
IsomorphismSimplifiedFpGroup (G). Inputs a finitely presented group “G”, forwhich GAP applies Tietze transformations to a copy in order to reduce it with re-spect to the number of generators, the number of relators, and the relator lengths.When we apply this to a finite-index subgroup of an oriented cocompact surfacegroup we get a one-relator group, as expected. This function returns an isomor-phism with domain G, codomain a group H isomorphic to G, so that the presenta-tion of H has been simplified using Tietze transformations.
IsOne (elem). Inputs an element “elem” from a group. Returns true if “elem” isequal to the identity, and false if “elem” is not. This function is not guaranteed toterminate.
Appendix B. Computer-assistance for the free group case
The code in §B.1 concretely defines maps φ and ψ from the proof of the freegroup case. Here K1 and K2 are the subgroups ∆ and ∆ , and K1.1 correspondsto the element A − in the free group h A, B i .Running this code outputs “false”. The GAP code verifies that each isomorphismexists and that the domains and ranges are all appropriate. In the end the GAPcode computes the function w = φ − ψφψ − φ − ψφψ − φ −
1N ABSTRACT COMMENSURATORS OF SURFACE GROUPS 9 with input Word =
BAB − A − . It outputs Word10 = A BAB − A , and verifiesthat these are not equal (although this last part is easily done by hand).After running the code below one can investigate properties of the map ψ . Forinstance, here ψ is evaluated on A − : gap> K1.1;A^-2gap> Image(psi, K1.1);A^-3 The variable K1.1, corresponding to the element A − in F , is shown here as beingmapped to A − .Moreover, one can verify that psi is an isomorphism K1 → K2 in GAP by checkingthat the map is surjective and that K1 and K2 have the same index in f: gap> Image(psi, K1) = K2;truegap> Index(f, K1) = Index(f, K2);true
B.1.
The code.
We ran the following code on GAP version 4.8.8. The usefulnessof the code below rests on the identifications of the generator K1.1 with A − and thegenerator K2.2 with A . Be aware that other versions of GAP may have differentimplementations of the function Kernel, in which A and A may correspond todifferent generators, or may not even be part of the chosen free generating set. Word := K1.2;; Word2 := Image(phi2, Word);;Word3 := Image(psi2, Word2);; Word4 := Image(phi, Word3);;Word5 := Image(psi, Word4);; Word6 := Image(phi2, Word5);;Word7 := Image(psi2, Word6);; Word8 := Image(phi, Word7);;Word9 := Image(psi, Word8);; Word10 := Image(phi2, Word9);;
Appendix C. Computer-assistance for the cocompact case
The code in §C.1 concretely defines maps φ and ψ from the proof of the co-compact Fuchsian case. Here K1 and K2 are the subgroups ∆ and ∆ , and K1.1corresponds to the element A − in the surface group g = h A, B, C, D : [
A, B ][ C, D ] i .In this case, more care must be taken in defining the map psi : K1 → K2 becauseK1 and K2 are not free groups. Explicitly defining maps from K1 to K2 using theGAP function GroupHomomorphismByImages usually results in maps that GAPcannot verify are well-defined isomorphisms (moreover, finding such maps fromscratch is prohibitively difficult). To get around this obstruction, we first simplifythe presentations of K1 and K2 before finding an isomorphism (akin to diagonalizinga matrix before doing computations).We briefly explain the construction of psi in the code now. First, the codedefines group homomorphisms Iso1 : K1 → fp1, Hom1 : fp1 → Image1, Iso2 : K1 → fp2, and Hom2 : fp2 → Image2. GAP does not view K1 and K2 as finitelypresented groups, and so the maps Iso1 and Iso2 are simply isomorphisms to theirfinite presentations (computed from the fact that K1 and K2 are finite-index normalsubgroups). The maps Hom1 and Hom2 are maps to simplified finite presentations.The resulting maps from this construction are version dependent, so running ourcode in a different version of GAP may result in lastMap not being well-defined(in which case GAP will throw a fault). The images of the maps Hom1 and Hom2have the following presentations. The red color indicates a part of the relation thatis significantly different from the rest.Image1 = h F , F , F , F , F , F , F , F , F , F , F , F , F , F :[ F , F ][ F , F ] F − F F [ F , F ][ F , F ] F − [ F , F ][ F , F ] = 1 i , Image2 = h F , F , F , F , F , F , F , F , F , F , F , F , F , F :[ F , F ] F [ F , F ][ F , F ] F − F − F [ F , F ][ F , F ][ F , F ] = 1 i . The image of A − under Hom1 ◦ Iso1 is F in Image1. Moreover, F maps to A under (Hom2 ◦ Iso2) − . Then lastMap : Image1 → Image2 is defined to be anisomorphism taking F to F . The resulting map psi : K1 → K2 is then defined bythe compositionpsi := flipHom ◦ Iso2 − ◦ Hom2 − ◦ lastMap ◦ Hom1 ◦ Iso1 , where flipHom sends A to A − , correcting that lastMap sends A − to A .After verifying that the map psi is well-defined, the computer uses the finitepresentation to verify that the input and output of the function w = φ − ψφψ − φ − ψφψ − φ −
1N ABSTRACT COMMENSURATORS OF SURFACE GROUPS 11 are not identical when evaluated at K1.2 = C in g (one can also do this last checkby hand using Dehn’s algorithm). The output to this evaluation, Word10, is: A^-1*B^2*A^-1*B^-2*(A^-1*C*D*C^-1*D^-1*A*B)^2*A*B^-2*A^3*B*A^-1*D*C*D^-1*C^-1*A*B^-1*A^-1*D*C*D^-1*C^-1*A^-2*B^2*A*B^-1*A^-1*D*C*D^-1*C*D*C^-1*D^-1*A*B*A^-1*B^-2*A^2*C*D*C^-1*D^-1*A*B*A^-1*C*D*C^-1*D^-1*A*B^-1*A^-3*B^2*A^-1*(B^-1*A^-1*D*C*D^-1*C^-1*A)^2*B^2*A*B^-2*A
As before, K1.1 corresponds to A − and applying ψ to it yields A − , as desired: gap> K1.1;A^-2gap> Image(psi, K1.1);A^-3 Moreover, one can verify that psi is an isomorphism K1 → K2 in GAP by checkingthat the map is surjective and that K1 and K2 have the same index in g: gap> Image(psi, K1) = K2;truegap> Index(g, K1) = Index(g, K2);true
C.1.
The code.
We ran the following code on GAP version 4.8.8. Note again thatdifferent versions of GAP may have different implementations of key functions,such as IsomorphismFpGroup and IsomorphismSimplifiedFpGroup, which are usedbelow to find explicit presentations of ∆ and ∆ . Because the definition of ψ usesthe presentations output by these functions, the definition below may not determinean isomorphism in other versions of GAP. Image1 := Image(Hom1);;Iso2 := IsomorphismFpGroup( K2 );;fp2 := Image(Iso2);;Hom2 := IsomorphismSimplifiedFpGroup(fp2);;Image2 := Image(Hom2);;
N ABSTRACT COMMENSURATORS OF SURFACE GROUPS 13
Word7 := Image(psi2, Word6);; Word8 := Image(phi, Word7);;Word9 := Image(psi, Word8);; Word10 := Image(phi2, Word9);;
References [Bau63] Gilbert Baumslag,
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