We argue that prejudice should be investigated in the context of social-cognitive development and the interplay between morality and group identity. Our new perspective examines how children consider group identity (and group norms) along with their developing moral beliefs about fairness and justice. This is achieved by developing an integrated framework drawing on developmental and social psychological theories of prejudice. This synthesis results in a perspective which provides a more contextualized analysis of prejudice development than previously offered by developmental theories.  We describe research which supports our view that social norms, intergroup contact and perceived out-group threat affect the relative weight children place on moral and group-based criteria during the development of prejudice

Abrams, Dominic

Killen, Melanie

Rutland, Adam

English

We show that the mapping class group (as well as closely related groups) of an orientable surface with finitely generated fundamental group has uniformly exponential growth. We further demonstrate the uniformly non-amenability of many of these groups. MSC 20F65 (primary), 20F38 (secondary) 1 Uniformly exponential growth The main purpose of this note is to demonstrate that the mapping class group of an orientable surface with finitely generated fundamental group has uniformly exponential growth. This result is new for surfaces of genus at least one, with the exception of the closed surface of genus two, and can be viewed as removing a possible avenue for showing that such mapping class groups are not linear. In this sense, our work is similar in spirit to the recent work of Brendle and Hamidi-Tehrani [4]. We go on to show that closely related groups of homotopy classes of homeomorphisms of surfaces, as well as analogous groups of automorphisms of free groups, also have uniformly exponential growth. We remark that, while the linearity of most surface mapping class groups is an open question

James W. Anderson

Javier Aramayona

Kenneth J. Shackleton

CiteSeerX

Uniformly exponential growth and mapping class groups of surfaces

Anderson, James W.

Aramayona, Javier

Shackleton, Kenneth J.

Infoscience - École polytechnique fédérale de Lausanne

Contemporary MathematicsUniformly exponential growth and mapping class groups ofsurfacesJ. W. Anderson, J. Aramayona, and K. J. ShackletonAbstract. We show that the mapping class group (as well as closely relatedgroups) of an orientable surface with finitely generated fundamental group hasuniformly exponential growth. We further demonstrate the uniformly non-amenability of many of these groups.1. Uniformly exponential growthThe main purpose of this note is to demonstrate that the mapping class groupof an orientable surface with finitely generated fundamental group has uniformlyexponential growth. This result is new for surfaces of genus at least one, with theexception of the closed surface of genus two, and can be viewed as removing apossible avenue for showing that such mapping class groups are not linear. In thissense, our work is similar in spirit to the recent work of Brendle and Hamidi-Tehrani[BH].We go on to show that closely related groups of homotopy classes of homeomor-phisms of surfaces, as well as analogous groups of automorphisms of free groups, alsohave uniformly exponential growth. We remark that, while the linearity of mostsurface mapping class groups is an open question, most automorphism groups of freegroups are known not to be linear. Specifically, as noted in Brendle and Hamidi-Tehrani [BH], it is known that Aut(Fn) is not linear for n ≥ 3 and Out(Fn) is notlinear for n ≥ 4, whereas both Aut(F2) and Out(F2) are linear. We also demon-strate the uniform non-amenability of many of the groups considered in this note;see Section 2.We begin by reviewing some basic definitions. For a survey of exponentialgrowth and uniformly exponential growth, we suggest the article of de la Harpe [H]and the references contained therein.Let G be a finitely generated group, and let S be a finite generating set forG. The length `S(g) of an element g ∈ G is the least integer k so that g can beexpressed as g = si1 · · · sik , where each sij ∈ S ∪ S−1. We define the length of theidentity element of G to be 0. LetBS(n) = {g ∈ G | `S(g) ≤ n}2000 Mathematics Subject Classification. MSC 20F65 (primary), 20F38 (secondary).c©0000 (copyright holder)12 J. W. ANDERSON, J. ARAMAYONA, AND K. J. SHACKLETONbe the ball of radius n about the identity element in G, and let |BS(n)| be thecardinality of BS(n).The exponential growth rate ω(G,S) of G with respect to S is defined to beω(G,S) = limn→∞n√|BS(n)|.(This limit exists, due to the submultiplicativity `S(gh) ≤ `S(g)`S(h) of the lengthfunction on G.) The group G has exponential growth if ω(G,S) > 1 for some (andhence for every) finite generating set S. Note that if G has a free subgroup of rank2, then G has exponential growth, though not conversely.We can also remove the dependence on particular generating sets by consideringω(G) = infSω(G,S),where the infimum is taken over all finite generating sets S of G. The group Gis said to have uniformly exponential growth if ω(G) > 1. For examples of groupsof exponential growth that do not have uniformly exponential growth, see Wilson[W].We make use of the following proposition from de la Harpe [H].Proposition 1 (from Proposition 2.3 of de la Harpe [H]). If G is a finitelygenerated group and if G′ is a quotient of G, then ω(G) ≥ ω(G′).We use the following result of Shalen and Wagreich [SW] to show the uni-formly exponential growth for certain groups related to the mapping class group(see Corollary 7 and Corollary 8).Lemma 2 (from Corollary 3.6 of Shalen andWagreich [SW]). Let G be a finitelygenerated group and let H be a finite index subgroup of G. If H has uniformlyexponential growth, then G has uniformly exponential growth.Remark 3. To the best of our knowledge, it is not yet known whether the con-verse of Lemma 2 holds. It seems the main difficulty lies in singling out an extendedgenerating set for G from one for H to give uniform embeddings on Cayley graphs.In this note, such issues represent only a minor inconvenience. A positive answerwould prove uniformly exponential growth to be a commensurability invariant; weconjecture that this should be true.Eskin, Mozes, and Oh [EMO] consider the question of uniformly exponentialgrowth for finitely generated subgroups of GL(n,C). In particular, they prove thefollowing.Theorem 4 (Corollary 1.4 of Eskin, Mozes, and Oh [EMO]). Let Γ be a finitelygenerated subgroup of GL(n,C). The following are equivalent:• Γ is not virtually nilpotent;• Γ is of uniformly exponential growth;• Γ is of exponential growth.The main tool we use is the following result, which should be viewed as animmediate corollary of Theorem 4.Theorem 5. Let G be a finitely generated group, and suppose that for somen ≥ 2 there exists a homomorphism ρ : G → GL(n,C) whose image ρ(G) is notvirtually nilpotent. Then, G has uniformly exponential growth.UNIFORMLY EXPONENTIAL GROWTH AND MAPPING CLASS GROUPS OF SURFACES 3Proof. Since ρ(G) is finitely generated and not virtually nilpotent, Theorem4 yields ω(ρ(G)) > 1 (that is, ρ(G) has uniformly exponential growth). Since ρ(G)is the homomorphic image of G, it is a quotient of G. Proposition 1 yields thatω(G) ≥ ω(ρ(G)) > 1, and so G has uniformly exponential growth. It is known that virtually nilpotent groups have polynomial growth (and con-versely, by work of Gromov [G]), whereas groups containing a free group of rankat least two have exponential growth.We are now ready to define the mapping class group of an orientable surfacewith finitely generated fundamental group. We cite the survey article of Ivanov [I]as our main reference for the mapping class group and its properties.Let Σ be a closed orientable surface of genus g ≥ 1, and let P be a finite set ofmarked points on Σ, where n = |P | ≥ 0. Themapping class groupM(Σ, P ) =Mg,nis the set of homotopy classes of orientation preserving homeomorphisms f : Σ→ Σfor which f(P ) = P , where the homotopies are required to keep each point of Pfixed. Note that the elements of Mg,n can permute the points of P . (We can alsoview Mg,n as the mapping class group of a surface of genus g with n punctures.)There is a natural surjective homomorphism fromMg,n to Symm(n), the sym-metric group on n letters, given by restricting the action ofMg,n to P . The kernelPMg,n of this homomorphism is the pure mapping class group, which is the sub-group of Mg,n fixing every element of P . Note that PMg,n is a subgroup of finiteindex in Mg,n.Now suppose n = |P | > 0 and let p ∈ P . By forgetting the marked point pwe find that every homeomorphism f : Σ→ Σ fixing P pointwise induces a home-omorphism f ′ : Σ → Σ fixing P \ {p} pointwise. In this way we get a surjectivehomomorphism PMg,n → PMg,n−1; in particular, we see that PMg,n homomor-phically surjects onto the mapping class group PMg,0 =Mg,0 =Mg of the surfaceΣ with no marked points.The extended mapping class group M±g,n is the group of homotopy classes of allhomeomorphisms of Σ fixing P setwise, and is a degree 2 extension of Mg,n.We can make similar definitions in the case that Σ is allowed to have non-empty boundary. Let Σ be a compact orientable surface of genus g ≥ 1 and letP be a finite set of n = |P | ≥ 0 marked points in the interior of Σ. Let mdenote the number of components of ∂Σ. The mapping class group Mg,n,m is thegroup of homotopy classes of orientation preserving homeomorphisms of Σ. Thepure mapping class group PMg,n,m is the subgroup of Mg,n,m consisting of thoseelements that permute neither the components of ∂Σ nor the elements of P . Asbefore, PMg,n,m is a subgroup of finite index in Mg,n,m. The extended mappingclass group M±g,n,m is the group of homotopy classes of all homeomorphisms of Σ,and is a degree 2 extension of Mg,n,m.We note that for a compact orientable surface Σ with non-empty boundary,there is a natural surjective homomorphism from PMg,n,m to PMg,n,0 = PMg,n,obtained by gluing discs to all the boundary components of Σ and extending thehomeomorphisms of Σ across these discs; this is discussed in detail in Theorem2.8.C of Ivanov [I].We are now ready for the main result of this note.Theorem 6. For g ≥ 1, the groups Mg and M±g have uniformly exponentialgrowth.4 J. W. ANDERSON, J. ARAMAYONA, AND K. J. SHACKLETONProof. Let Σ be a closed surface of genus g ≥ 1 and consider the mappingclass group Mg of Σ. Recall that Mg is finitely generated. We show the result forMg; Lemma 2 then yields the result for M±g , which is a degree 2 extension of Mgas we noted above. The natural action of Mg on H1(Σ,Z) = pi1(Σ)/[pi1(Σ), pi1(Σ)]yields a surjective homomorphism ρ : Mg → Sp(2g,Z). Since Sp(2g,Z) containsSp(2,Z) ∼= SL(2,Z) as a subgroup and since SL(2,Z) contains a Z ∗ Z subgroup,we see that ρ(Mg) = Sp(2g,Z) is not virtually nilpotent. Hence, by Theorem 5,we see that Mg has uniformly exponential growth; by Lemma 2, we see that M±ghas uniformly exponential growth.[To see that Sp(2g,Z) contains Sp(2,Z) ∼= SL(2,Z), recall that Sp(2g,Z) isdefined to be the group of 2g × 2g-matrices preserving a non-degenerate, skew-symmetric bilinear form. If we take the form q(x,y) = x1y2−x2y1+· · ·+x2g−1y2g−x2gy2g−1, then any block diagonal matrix of the form(A 02,2g−202g−2,2 I2g−2)preserves q and hence lies in Sp(2g,Z), where A lies in the group Sp(2,Z) preservingthe quadratic form q0(x,y) = x1y2 − x2y1, 0s,t is the s × t matrix of zeroes, andIk is the k × k identity matrix.]We saw before that there is a surjective homomorphism from PMg,n to PMg =Mg, and that PMg,n has finite index inMg,n. Further,Mg,n is finitely generated.Proposition 1 and Lemma 2 then yield the following corollary.Corollary 7. For g ≥ 1 and n > 0, the groups PMg,n, Mg,n and M±g,n haveuniformly exponential growth.We also saw that, in the case of surfaces with boundary, there is a homomor-phism from PMg,n,m to PMg,n, and that PMg,n,m has finite index in Mg,n,m.Further,Mg,n,m is finitely generated. We thus get the following result, again usingProposition 1 and Lemma 2 .Corollary 8. For g ≥ 1, m > 0, and n ≥ 0, the groups PMg,n,m, Mg,n,mand M±g,n,m have uniformly exponential growth.For g = 0, the methods we use here do not apply. We note that when n ≥ 4,it is a remarkable result of Bigelow [Bi] and Krammer [Kr] that each M0,n islinear (and not virtually nilpotent) and hence has uniformly exponential growth byTheorem 4. For g = 2 and n = 0, it is a result of Bigelow and Budney [BB] andof Korkmaz [Ko] that M2,0 is linear and hence has uniformly exponential growth,again by Theorem 4.2. Uniform non-amenabilityWe note that the same argument we have given for the uniformly exponentialgrowth of the mapping class groups Mg also yields their uniform non-amenability.Following Arzhantseva et al. [ABLRSV], given a finitely generated group G withfinite generating set X , define the inner boundary of a finite subset A of G to be∂X(A) = {a ∈ A | ax 6∈ A for some x ∈ X ∪X−1}.UNIFORMLY EXPONENTIAL GROWTH AND MAPPING CLASS GROUPS OF SURFACES 5The uniform Følner constant of G is thenFol(G) = infXinfA|∂X(A)||A|,where the outer infimum is taken over all (finite) generating sets X of G and theinner infimum is taken over all finite subsets A of G. We say that G is uniformlynon-amenable if Fol(G) > 0.We make use of the following results about uniform non-amenability, which arethe analogues of the similar results for uniformly exponential growth. The firstnotes how uniform non-amenability behaves under quotients.Proposition 9 (from Theorem 4.1 of Arzhantseva et al. [ABLRSV]). LetG be a finitely generated group, and let N be a normal subgroup of G. Then,Fol(G) ≥ Fol(G/N).The second describes the uniform non-amenability of linear groups.Theorem 10 (from Corollary 1.2 of Breuillard and Gelander [BrG]). Let Γ bea finitely generated subgroup of GL(n,C). Assume that Γ is not amenable. Then,Γ is uniformly non-amenable.We now consider the uniform non-amenability of mapping class groups.Theorem 11. The groups Mg are uniformly non-amenable for all g ≥ 1.Proof. Proposition 9 states that if G′ is a quotient of the finitely generatedgroup G, then Fol(G) ≥ Fol(G′); this takes the place of Proposition 1. Theorem10 states that finitely generated non-virtually nilpotent linear groups over any fieldare uniformly non-amenable; this takes the place of Theorem 4, and the analogueof Theorem 5 for uniform non-amenability then follows immediately. The proof ofTheorem 6 then proceeds to show that the groupsMg are uniformly non-amenablefor all g ≥ 1. Applying Proposition 9 then shows that any groups that surject onto one of theMg for g ≥ 1 are also uniformly non-amenable; see Corollary 7 and Corollary 8.This leaves outM±g , as the analogue of Lemma 2 is not known to hold for uniformnon-amenability.3. Automorphisms of free groupsAnalogous results hold for the automorphism group Aut(Fn) of the free groupFn of rank n and for the outer automorphism group Out(Fn), the quotient ofAut(Fn) by the group of inner automorphisms.Theorem 12. For n ≥ 2, the groups Aut(Fn) and Out(Fn) have uniformlyexponential growth.Proof. Begin with Out(Fn). The natural action of Out(Fn) on Fn/[Fn, Fn] =Zn yields a surjective homomorphism ρ : Out(Fn) → GL(n,Z). Since GL(n,Z)is finitely generated and is not virtually nilpotent (as SL(2,Z) ⊂ SL(n,Z) ⊂GL(n,Z)), Theorem 5 yields that Out(Fn) has uniformly exponential growth.That Aut(Fn) has uniformly exponential growth follows from Proposition 1,since Out(Fn) is a quotient of Aut(Fn). 6 J. W. ANDERSON, J. ARAMAYONA, AND K. J. SHACKLETONAs above, the same proof (with appropriate substitutions, as in the proof ofTheorem 11) shows that for n ≥ 2, the groups Aut(Fn) and Out(Fn) are uniformlynon-amenable.Acknowledgements. Parts of this paper were written during visits of thesecond author to the Bernoulli Center, Lausanne, and the third author to theTokyo Institute of Technology. Both authors wish to express their gratitude tothese institutions. The third author was supported by an EPSRC studentship.The first author would also like to thank Gareth Jones for helpful conversations.References[ABLRSV] G. N. Arzhantseva, J. Burillo, M. Lustig, L. Reeves, H. Short, E. Ventura, Uniformnon-amenability, Adv. Math. 197 (2005), 499–522.[Bi] S. J. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001), 471–486.[BB] S. J. Bigelow, R. D. Budney, The mapping class group of a genus two surface is linear,Algebr. Geom. Topol. 1 (2001), 699–708.[BH] T. E. Brendle, H. Hamidi-Tehrani, On the linearity problem for mapping class groups,Algebr. Geom. Topol. 1 (2001), 445–468.[BrG] E. Breuillard, T. Gelander, Cheeger constant and algebraic entropy of linear groups, Int.Math. Res. Not. 56 (2005), 3511–3523.[EMO] A. Eskin, S. Mozes, H. Oh, On uniform exponential growth for linear groups, Invent.Math. 160 (2005), 1–30.[H] P. de la Harpe, Uniform Growth in Groups of Exponential Growth, Geom. Dedicata 9 (2000),1–17.[G] M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Etudes Sci.Publ. Math. 53 (1981), 53–73.[I] N. V. Ivanov, Mapping class groups, in Handbook of Geometric Topology, edited by R. J.Daverman and R. B. Sher, Elsevier Science B. V., London, 2002, 523–633.[Ko] M. Korkmaz, On the linearity of certain mapping class groups, Turkish J. Math. 24 (2000),367–371.[Kr] D. Krammer, Braid groups are linear, Ann. of Math. 155 (2002), 131–156.[SW] P. B. Shalen and P. Wagreich, Growth Rates, Zp-Homology, and Volumes of Hyperbolic3-Manifolds, Trans. Amer. Math. Soc. 331 (1992), 895–917.[W] J. S. Wilson, On exponential growth and uniformly exponential growth for groups, Invent.Math. 155 (2004), 287–303.School of Mathematics, University of Southampton, Southampton SO17 1BJ, Eng-landE-mail address: j.w.anderson@soton.ac.ukMathematics Institute, University of Warwick, Coventry CV4 7AL, EnglandE-mail address: jaram@maths.warwick.ac.ukDepartment of Mathematical and Computing Sciences, Tokyo Institute of Tech-nology, 2-12-1 O-okayama, Meguro-ku, Tokyo, 152-8552 JapanE-mail address: kjs2006@alumni.soton.ac.uk

Uniformly Exponential Growth and Mapping Class Groups of Surfaces

J. W. Anderson

J. Aramayona

K. J. Shackleton

Crossref

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A new social-cognitive developmental perspective on prejudice: The interplay between morality and group identity.

We argue that prejudice should be investigated in the context of social-cognitive development and the interplay between morality and group identity. Our new perspective examines how children consider group identity (and group norms) along with their developing moral beliefs about fairness and justice. This is achieved by developing an integrated framework drawing on developmental and social psychological theories of prejudice. This synthesis results in a perspective that provides a more contextualized analysis of prejudice development than that previously offered by developmental theories. We describe research that supports our view that social norms, intergroup contact, and perceived outgroup threat affect the relative weight children place on moral and group-based criteria during the development of prejudice

Goldsmiths Research Online

A New Social-Cognitive Developmental Perspective on Prejudice: The Interplay Between Morality and Group Identity

We show that the mapping class group (as well as closely related groups) of an orientable surface with finitely generated fundamental group has uniformly exponential growth. We further demonstrate the uniformly non-amenability of many of these groups

Anderson, J.W.

Aramayona, J.

Shackleton, K.J.

Southampton (e-Prints Soton)

Contemporary MathematicsUniformly exponential growth and mapping class groups ofsurfacesJ. W. Anderson, J. Aramayona, and K. J. ShackletonAbstract. We show that the mapping class group (as well as closely relatedgroups) of an orientable surface with ﬁnitely generated fundamental group hasuniformly exponential growth. We further demonstrate the uniformly non-amenability of many of these groups.1. Uniformly exponential growthThe main purpose of this note is to demonstrate that the mapping class groupof an orientable surface with ﬁnitely generated fundamental group has uniformlyexponential growth. This result is new for surfaces of genus at least one, with theexception of the closed surface of genus two, and can be viewed as removing apossible avenue for showing that such mapping class groups are not linear. In thissense, our work is similar in spirit to the recent work of Brendle and Hamidi-Tehrani[BH].We go on to show that closely related groups of homotopy classes of homeomor-phisms of surfaces, as well as analogous groups of automorphisms of free groups, alsohave uniformly exponential growth. We remark that, while the linearity of mostsurface mapping class groups is an open question, most automorphism groups of freegroups are known not to be linear. Speciﬁcally, as noted in Brendle and Hamidi-Tehrani [BH], it is known that Aut(Fn) is not linear for n ≥ 3 and Out(Fn) is notlinear for n ≥ 4, whereas both Aut(F2) and Out(F2) are linear. We also demon-strate the uniform non-amenability of many of the groups considered in this note;see Section 2.We begin by reviewing some basic deﬁnitions. For a survey of exponentialgrowth and uniformly exponential growth, we suggest the article of de la Harpe [H]and the references contained therein.Let G be a ﬁnitely generated group, and let S be a ﬁnite generating set forG. The length `S(g) of an element g ∈ G is the least integer k so that g can beexpressed as g = si1 ···sik, where each sij ∈ S ∪ S−1. We deﬁne the length of theidentity element of G to be 0. LetBS(n) = {g ∈ G | `S(g) ≤ n}2000 Mathematics Subject Classiﬁcation. MSC 20F65 (primary), 20F38 (secondary).c ￿0000 (copyright holder)12 J. W. ANDERSON, J. ARAMAYONA, AND K. J. SHACKLETONbe the ball of radius n about the identity element in G, and let |BS(n)| be thecardinality of BS(n).The exponential growth rate ω(G,S) of G with respect to S is deﬁned to beω(G,S) = limn→∞n p|BS(n)|.(This limit exists, due to the submultiplicativity `S(gh) ≤ `S(g)`S(h) of the lengthfunction on G.) The group G has exponential growth if ω(G,S) > 1 for some (andhence for every) ﬁnite generating set S. Note that if G has a free subgroup of rank2, then G has exponential growth, though not conversely.We can also remove the dependence on particular generating sets by consideringω(G) = infSω(G,S),where the inﬁmum is taken over all ﬁnite generating sets S of G. The group Gis said to have uniformly exponential growth if ω(G) > 1. For examples of groupsof exponential growth that do not have uniformly exponential growth, see Wilson[W].We make use of the following proposition from de la Harpe [H].Proposition 1 (from Proposition 2.3 of de la Harpe [H]). If G is a ﬁnitelygenerated group and if G0 is a quotient of G, then ω(G) ≥ ω(G0).We use the following result of Shalen and Wagreich [SW] to show the uni-formly exponential growth for certain groups related to the mapping class group(see Corollary 7 and Corollary 8).Lemma 2 (from Corollary 3.6 of Shalen and Wagreich [SW]). Let G be a ﬁnitelygenerated group and let H be a ﬁnite index subgroup of G. If H has uniformlyexponential growth, then G has uniformly exponential growth.Remark 3. To the best of our knowledge, it is not yet known whether the con-verse of Lemma 2 holds. It seems the main diﬃculty lies in singling out an extendedgenerating set for G from one for H to give uniform embeddings on Cayley graphs.In this note, such issues represent only a minor inconvenience. A positive answerwould prove uniformly exponential growth to be a commensurability invariant; weconjecture that this should be true.Eskin, Mozes, and Oh [EMO] consider the question of uniformly exponentialgrowth for ﬁnitely generated subgroups of GL(n,C). In particular, they prove thefollowing.Theorem 4 (Corollary 1.4 of Eskin, Mozes, and Oh [EMO]). Let Γ be a ﬁnitelygenerated subgroup of GL(n,C). The following are equivalent:• Γ is not virtually nilpotent;• Γ is of uniformly exponential growth;• Γ is of exponential growth.The main tool we use is the following result, which should be viewed as animmediate corollary of Theorem 4.Theorem 5. Let G be a ﬁnitely generated group, and suppose that for somen ≥ 2 there exists a homomorphism ρ : G → GL(n,C) whose image ρ(G) is notvirtually nilpotent. Then, G has uniformly exponential growth.UNIFORMLY EXPONENTIAL GROWTH AND MAPPING CLASS GROUPS OF SURFACES 3Proof. Since ρ(G) is ﬁnitely generated and not virtually nilpotent, Theorem4 yields ω(ρ(G)) > 1 (that is, ρ(G) has uniformly exponential growth). Since ρ(G)is the homomorphic image of G, it is a quotient of G. Proposition 1 yields thatω(G) ≥ ω(ρ(G)) > 1, and so G has uniformly exponential growth. ￿It is known that virtually nilpotent groups have polynomial growth (and con-versely, by work of Gromov [G]), whereas groups containing a free group of rankat least two have exponential growth.We are now ready to deﬁne the mapping class group of an orientable surfacewith ﬁnitely generated fundamental group. We cite the survey article of Ivanov [I]as our main reference for the mapping class group and its properties.Let Σ be a closed orientable surface of genus g ≥ 1, and let P be a ﬁnite set ofmarked points on Σ, where n = |P| ≥ 0. The mapping class group M(Σ,P) = Mg,nis the set of homotopy classes of orientation preserving homeomorphisms f : Σ → Σfor which f(P) = P, where the homotopies are required to keep each point of Pﬁxed. Note that the elements of Mg,n can permute the points of P. (We can alsoview Mg,n as the mapping class group of a surface of genus g with n punctures.)There is a natural surjective homomorphism from Mg,n to Symm(n), the sym-metric group on n letters, given by restricting the action of Mg,n to P. The kernelPMg,n of this homomorphism is the pure mapping class group, which is the sub-group of Mg,n ﬁxing every element of P. Note that PMg,n is a subgroup of ﬁniteindex in Mg,n.Now suppose n = |P| > 0 and let p ∈ P. By forgetting the marked point pwe ﬁnd that every homeomorphism f : Σ → Σ ﬁxing P pointwise induces a home-omorphism f0 : Σ → Σ ﬁxing P \ {p} pointwise. In this way we get a surjectivehomomorphism PMg,n → PMg,n−1; in particular, we see that PMg,n homomor-phically surjects onto the mapping class group PMg,0 = Mg,0 = Mg of the surfaceΣ with no marked points.The extended mapping class group M±g,n is the group of homotopy classes of allhomeomorphisms of Σ ﬁxing P setwise, and is a degree 2 extension of Mg,n.We can make similar deﬁnitions in the case that Σ is allowed to have non-empty boundary. Let Σ be a compact orientable surface of genus g ≥ 1 and letP be a ﬁnite set of n = |P| ≥ 0 marked points in the interior of Σ. Let mdenote the number of components of ∂Σ. The mapping class group Mg,n,m is thegroup of homotopy classes of orientation preserving homeomorphisms of Σ. Thepure mapping class group PMg,n,m is the subgroup of Mg,n,m consisting of thoseelements that permute neither the components of ∂Σ nor the elements of P. Asbefore, PMg,n,m is a subgroup of ﬁnite index in Mg,n,m. The extended mappingclass group M±g,n,m is the group of homotopy classes of all homeomorphisms of Σ,and is a degree 2 extension of Mg,n,m.We note that for a compact orientable surface Σ with non-empty boundary,there is a natural surjective homomorphism from PMg,n,m to PMg,n,0 = PMg,n,obtained by gluing discs to all the boundary components of Σ and extending thehomeomorphisms of Σ across these discs; this is discussed in detail in Theorem2.8.C of Ivanov [I].We are now ready for the main result of this note.Theorem 6. For g ≥ 1, the groups Mg and M±g have uniformly exponentialgrowth.4 J. W. ANDERSON, J. ARAMAYONA, AND K. J. SHACKLETONProof. Let Σ be a closed surface of genus g ≥ 1 and consider the mappingclass group Mg of Σ. Recall that Mg is ﬁnitely generated. We show the result forMg; Lemma 2 then yields the result for M±g , which is a degree 2 extension of Mgas we noted above. The natural action of Mg on H1(Σ,Z) = π1(Σ)/[π1(Σ),π1(Σ)]yields a surjective homomorphism ρ : Mg → Sp(2g,Z). Since Sp(2g,Z) containsSp(2,Z) ∼ = SL(2,Z) as a subgroup and since SL(2,Z) contains a Z ∗ Z subgroup,we see that ρ(Mg) = Sp(2g,Z) is not virtually nilpotent. Hence, by Theorem 5,we see that Mg has uniformly exponential growth; by Lemma 2, we see that M±ghas uniformly exponential growth.[To see that Sp(2g,Z) contains Sp(2,Z) ∼ = SL(2,Z), recall that Sp(2g,Z) isdeﬁned to be the group of 2g × 2g-matrices preserving a non-degenerate, skew-symmetric bilinear form. If we take the form q(x,y) = x1y2−x2y1+···+x2g−1y2g−x2gy2g−1, then any block diagonal matrix of the form￿A 02,2g−202g−2,2 I2g−2￿preserves q and hence lies in Sp(2g,Z), where A lies in the group Sp(2,Z) preservingthe quadratic form q0(x,y) = x1y2 − x2y1, 0s,t is the s × t matrix of zeroes, andIk is the k × k identity matrix.]￿We saw before that there is a surjective homomorphism from PMg,n to PMg =Mg, and that PMg,n has ﬁnite index in Mg,n. Further, Mg,n is ﬁnitely generated.Proposition 1 and Lemma 2 then yield the following corollary.Corollary 7. For g ≥ 1 and n > 0, the groups PMg,n, Mg,n and M±g,n haveuniformly exponential growth.We also saw that, in the case of surfaces with boundary, there is a homomor-phism from PMg,n,m to PMg,n, and that PMg,n,m has ﬁnite index in Mg,n,m.Further, Mg,n,m is ﬁnitely generated. We thus get the following result, again usingProposition 1 and Lemma 2 .Corollary 8. For g ≥ 1, m > 0, and n ≥ 0, the groups PMg,n,m, Mg,n,mand M±g,n,m have uniformly exponential growth.For g = 0, the methods we use here do not apply. We note that when n ≥ 4,it is a remarkable result of Bigelow [Bi] and Krammer [Kr] that each M0,n islinear (and not virtually nilpotent) and hence has uniformly exponential growth byTheorem 4. For g = 2 and n = 0, it is a result of Bigelow and Budney [BB] andof Korkmaz [Ko] that M2,0 is linear and hence has uniformly exponential growth,again by Theorem 4.2. Uniform non-amenabilityWe note that the same argument we have given for the uniformly exponentialgrowth of the mapping class groups Mg also yields their uniform non-amenability.Following Arzhantseva et al. [ABLRSV], given a ﬁnitely generated group G withﬁnite generating set X, deﬁne the inner boundary of a ﬁnite subset A of G to be∂X(A) = {a ∈ A | ax 6∈ A for some x ∈ X ∪ X−1}.UNIFORMLY EXPONENTIAL GROWTH AND MAPPING CLASS GROUPS OF SURFACES 5The uniform Følner constant of G is thenFol(G) = infXinfA|∂X(A)||A|,where the outer inﬁmum is taken over all (ﬁnite) generating sets X of G and theinner inﬁmum is taken over all ﬁnite subsets A of G. We say that G is uniformlynon-amenable if Fol(G) > 0.We make use of the following results about uniform non-amenability, which arethe analogues of the similar results for uniformly exponential growth. The ﬁrstnotes how uniform non-amenability behaves under quotients.Proposition 9 (from Theorem 4.1 of Arzhantseva et al. [ABLRSV]). LetG be a ﬁnitely generated group, and let N be a normal subgroup of G. Then,Fol(G) ≥ Fol(G/N).The second describes the uniform non-amenability of linear groups.Theorem 10 (from Corollary 1.2 of Breuillard and Gelander [BrG]). Let Γ bea ﬁnitely generated subgroup of GL(n,C). Assume that Γ is not amenable. Then,Γ is uniformly non-amenable.We now consider the uniform non-amenability of mapping class groups.Theorem 11. The groups Mg are uniformly non-amenable for all g ≥ 1.Proof. Proposition 9 states that if G0 is a quotient of the ﬁnitely generatedgroup G, then Fol(G) ≥ Fol(G0); this takes the place of Proposition 1. Theorem10 states that ﬁnitely generated non-virtually nilpotent linear groups over any ﬁeldare uniformly non-amenable; this takes the place of Theorem 4, and the analogueof Theorem 5 for uniform non-amenability then follows immediately. The proof ofTheorem 6 then proceeds to show that the groups Mg are uniformly non-amenablefor all g ≥ 1. ￿Applying Proposition 9 then shows that any groups that surject onto one of theMg for g ≥ 1 are also uniformly non-amenable; see Corollary 7 and Corollary 8.This leaves out M±g , as the analogue of Lemma 2 is not known to hold for uniformnon-amenability.3. Automorphisms of free groupsAnalogous results hold for the automorphism group Aut(Fn) of the free groupFn of rank n and for the outer automorphism group Out(Fn), the quotient ofAut(Fn) by the group of inner automorphisms.Theorem 12. For n ≥ 2, the groups Aut(Fn) and Out(Fn) have uniformlyexponential growth.Proof. Begin with Out(Fn). The natural action of Out(Fn) on Fn/[Fn,Fn] =Zn yields a surjective homomorphism ρ : Out(Fn) → GL(n,Z). Since GL(n,Z)is ﬁnitely generated and is not virtually nilpotent (as SL(2,Z) ⊂ SL(n,Z) ⊂GL(n,Z)), Theorem 5 yields that Out(Fn) has uniformly exponential growth.That Aut(Fn) has uniformly exponential growth follows from Proposition 1,since Out(Fn) is a quotient of Aut(Fn). ￿6 J. W. ANDERSON, J. ARAMAYONA, AND K. J. SHACKLETONAs above, the same proof (with appropriate substitutions, as in the proof ofTheorem 11) shows that for n ≥ 2, the groups Aut(Fn) and Out(Fn) are uniformlynon-amenable.Acknowledgements. Parts of this paper were written during visits of thesecond author to the Bernoulli Center, Lausanne, and the third author to theTokyo Institute of Technology. Both authors wish to express their gratitude tothese institutions. The third author was supported by an EPSRC studentship.The ﬁrst author would also like to thank Gareth Jones for helpful conversations.References[ABLRSV] G. N. Arzhantseva, J. Burillo, M. Lustig, L. Reeves, H. Short, E. Ventura, Uniformnon-amenability, Adv. Math. 197 (2005), 499–522.[Bi] S. J. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001), 471–486.[BB] S. J. Bigelow, R. D. Budney, The mapping class group of a genus two surface is linear,Algebr. Geom. Topol. 1 (2001), 699–708.[BH] T. E. Brendle, H. Hamidi-Tehrani, On the linearity problem for mapping class groups,Algebr. Geom. Topol. 1 (2001), 445–468.[BrG] E. Breuillard, T. Gelander, Cheeger constant and algebraic entropy of linear groups, Int.Math. Res. Not. 56 (2005), 3511–3523.[EMO] A. Eskin, S. Mozes, H. Oh, On uniform exponential growth for linear groups, Invent.Math. 160 (2005), 1–30.[H] P. de la Harpe, Uniform Growth in Groups of Exponential Growth, Geom. Dedicata 9 (2000),1–17.[G] M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Etudes Sci.Publ. Math. 53 (1981), 53–73.[I] N. V. Ivanov, Mapping class groups, in Handbook of Geometric Topology, edited by R. J.Daverman and R. B. Sher, Elsevier Science B. V., London, 2002, 523–633.[Ko] M. Korkmaz, On the linearity of certain mapping class groups, Turkish J. Math. 24 (2000),367–371.[Kr] D. Krammer, Braid groups are linear, Ann. of Math. 155 (2002), 131–156.[SW] P. B. Shalen and P. Wagreich, Growth Rates, Zp-Homology, and Volumes of Hyperbolic3-Manifolds, Trans. Amer. Math. Soc. 331 (1992), 895–917.[W] J. S. Wilson, On exponential growth and uniformly exponential growth for groups, Invent.Math. 155 (2004), 287–303.School of Mathematics, University of Southampton, Southampton SO17 1BJ, Eng-landE-mail address: j.w.anderson@soton.ac.ukMathematics Institute, University of Warwick, Coventry CV4 7AL, EnglandE-mail address: jaram@maths.warwick.ac.ukDepartment of Mathematical and Computing Sciences, Tokyo Institute of Tech-nology, 2-12-1 O-okayama, Meguro-ku, Tokyo, 152-8552 JapanE-mail address: kjs2006@alumni.soton.ac.uk

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