Harmonic functions of linear growth on solvable groups
aa r X i v : . [ m a t h . G R ] O c t HARMONIC FUNCTIONS OF LINEAR GROWTH ONSOLVABLE GROUPS
TOM MEYEROVITCH AND ARIEL YADIN
Abstract.
In this work we study the structure of finitely generated groupsfor which a space of harmonic functions with fixed polynomial growth is finitedimensional. It is conjectured that such groups must be virtually nilpotent (theconverse direction to Kleiner’s theorem). We prove that this is indeed the casefor solvable groups. The investigation is partly motivated by Kleiner’s proof forGromov’s theorem on groups of polynomial growth.
Contents
1. Introduction 12. Further research directions and open questions 83. Reduction from solvable to subgroups of A ( F ) 114. Random walks on the reals 185. A positive harmonic function for subgroups of A ( F ) 22References 361. Introduction
Based on Colding and Minicozzi’s solution to Yau’s Conjecture [9], in 2007Kleiner proved the following theorem [20]: For any finitely generated group G of ∗ Ben Gurion University of the Negev. email: { mtom, yadina } @math.bgu.ac.il .T.M. would like to acknowledge funding from the People Programme (Marie Curie Actions)of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grantagreement no. 333598, and from the Israel Science Foundation (grant no. 626/14). polynomial growth, the space of harmonic functions on G of some fixed polynomialgrowth is a finite dimensional vector space. Using this theorem Kleiner obtaineda non-trivial finite dimensional representation of G , and discovered a new proof ofGromov’s theorem [14]: Any finitely generated group of polynomial growth has afinite index subgroup that is nilpotent ( i.e. is virtually nilpotent).A natural question along these lines is whether the converse of Kleiner’s theoremholds. That is: Conjecture 1.1.
Let G be a finitely generated group, and let µ be a symmetricprobability measure on G , with finite support that generates G . Let HF k ( G , µ ) denote the space of µ -harmonic functions on G whose growth is bounded by adegree k polynomial.Then the following are equivalent: (1) G is virtually nilpotent. (2) G has polynomial growth. (3) dim HF k ( G , µ ) < ∞ for all k . (4) There exists k ≥ such that dim HF k ( G , µ ) < ∞ . Let us consider the space of bounded harmonic functions ( i.e. HF ), which wealso denote by BHF = BHF ( G , µ ). The space BHF ( G , µ ) is isomorphic to the spaceof bounded functions on the Poisson-Furstenberg boundary of ( G , µ ). This objecthas been studied extensively in the literature over the past. We refer to the seminalpaper [19] of Ka˘ımanovich and Vershik also [12, 13] for background and more onthis object. From the theory of Poisson-Furstenberg boundaries it follows thatwhen the Poisson-Furstenberg boundary is not one point, then it must be infinite.So if the dimension of the bounded harmonic functions on G is finite, then thePoisson-Furstenberg boundary is finite, and thus trivial (a singleton). Hence, theonly bounded harmonic functions in this case are the constants. Positive harmonic ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 3 functions (Martin boundary, see e.g. [25]) have also been extensively studied inthe literature.Recently, there has been growing interest in the study of spaces of unboundedharmonic functions on groups and other homogeneous spaces (see e.g. [4] andreferences therein). As mentioned, Kleiner [20] used the space HF of linearlygrowing functions to produce a finite dimensional representation for groups ofpolynomial growth, which lead to a new proof of Gromov’s theorem.The main result of this paper, is a proof of Conjecture 1.1 in the case where G is a solvable group. Our proofs and tools are probabilistic utilizing the theory ofrandom walks on groups and martingales.One consequence of our results is a structure theorem for the space of linearlygrowing harmonic functions for general groups where this space is finite dimen-sional: Up to an additive constant and passing to a finite index subgroups any suchfunction must be a homomophism into the additive group R . In a follow-up paperjoint with Idan Perl and Matthew Tointon [22] we provide, along with additionalresults, a structure theorem for the space of harmonic functions of polynomialgrowth (in the finite-dimensional case).After introducing some notation, we will precisely state the main contributionsof this work in Section 1.2. Acknowledgements:
The birth of this work was during a research seminar inBen Gurion University. We thank the participants of the this seminar for theirpart. We acknowledge interesting conversations, encouragement and valuable sug-gestions from Amichai Aisenmann, Tsachik Gelander, Yair Glasner, Gady Kozma,Yehuda Shalom, Maud Szusterman and the anonymous referees.1.1.
Notation.
Throughout, G is a countable group generated by a finite sym-metric set; G = h S i , S = S − , | S | < ∞ , and µ is a probability measure on G . Thegenerating set induces a metric dist S on G , namely the graph metric of the Cayleygraph with respect to S . This metric is invariant to the action of G from the left. TOM MEYEROVITCH AND ARIEL YADIN
We write | x | = | x | S = dist S ( x,
1) for x ∈ G . The metrics obtained by differentchoices of generating sets are bi-Lipshcitz.The pair ( G , µ ) is called a measured group . We will always assume µ is a symmetric probability measure on G ; i.e. µ ( x ) = µ ( x − ) for all x ∈ G , and that µ is adapted : the support of µ generates G . We say that µ is smooth if thegenerating function ϕ µ ( ζ ) := P x µ ( x ) e ζ | x | has positive radius of convergence.We say that µ is courteous if it is a symmetric adapted and smooth probabilitymeasure.Section 3.1 details a bit the properties of smooth measures and explains whythe class of courteous measures is a natural class of probability measures to workwith.Clearly, any measure µ with finite support is smooth. A primary example fora courteous measure is the measure µ = | S | P s ∈ S δ s , distributed uniformly over afinite, symmetric generating set S ⊂ G .A µ -harmonic function f : G → C is a function such that for all x ∈ G , f ∗ µ ( x ) := P s f ( xs − ) µ ( s ) = f ( x ).A group G acts naturally on functions on the group; namely by xf ( y ) = f ( x − y ).The space BHF = BHF ( G , µ ) of bounded harmonic functions is a G -invariantspace; that is G BHF = BHF .A measured group ( G , µ ) with the property that all bounded harmonic functionsare constant is called Liouville . The property of having a finite dimensional spaceof µ -harmonic functions of linear growth can be viewed as a refinement of theLiouville property.We recall the following fact about bounded harmonic functions: Lemma 1.2.
The only non-trivial finite-dimensional G -invariant subspace of BHF ( G , µ ) is the constant functions; that is if V ≤ BHF is G -invariant and dim V < ∞ then V = C .In particular, if dim BHF < ∞ then ( G , µ ) is Liouville. ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 5
Lemma 1.2 above is relatively classical in the study of Poisson-Furstenbergboundaries. In private communication, Yehuda Shalom presented a slick argumentbased on the maximum principle and a compactness argument: The orbit closure ofa non-trivial harmonic function with finite dimensional orbit contains a harmonicfunction with a proper maximum.For a function f : G → C define the polynomial- k -semi-norm:(1) || f || k := lim sup r →∞ r − k · max | x |≤ r | f ( x ) | . We denote(2) HF k ( G , µ ) := (cid:8) f : G → C (cid:12)(cid:12) k f k k < ∞ , f is µ -harmonic (cid:9) . The space HF k = HF k ( G , µ ) is the space of µ -harmonic functions with polyno-mial growth of degree at most k . Note that || xf || k = || f || k for any x ∈ G , f ∈ C G ,so the space HF k is G -invariant. The space HF k ( G , µ ) does not depend on thechoice of generating set for G (but does inherently depend on the measure µ ).Recall that a countable group G is amenable if for any K ⊂ G and any ǫ > F ⊂ G so that | KF | ≤ (1 + ǫ ) | F | . There are numerousequivalent definitions of amenability. For definitions and a detailed account ofamenability for locally compact groups we refer for instance to [23]. Proposition 1.3. If dim HF k ( G , µ ) < ∞ for some k ≥ , then G is amenable.Proof. By Lemma 1.2, if ( G , µ ) is not Liouville then dim BHF ( G , µ ) = ∞ .Since BHF ( G , µ ) = HF ( G , µ ) is a subspace of HF k ( G , µ ) for all k ≥
0, it followsthat the assumption dim HF k ( G , µ ) < ∞ implies that G is Liouville.Rosenblatt [24] and independently Ka˘ımanovich-Vershik [19] showed that if( G , µ ) is Liouville then G is amenable. ⊓⊔ By Proposition 1.3, it suffices to consider amenable groups in Conjecture 1.1.
TOM MEYEROVITCH AND ARIEL YADIN A random walk on G with step distribution µ is a random sequence ( X t ) t defined by X t = X S S · · · S t , where ( S t ) t are i.i.d.- µ . The canonical measureand expectation on G N of this process, conditioned on X = x , are denoted P x , E x .When the subscript is omitted we mean P = P , E = E (1 = 1 G is the unit elementin G ).Random walks and harmonic functions are intimately related. f is µ -harmonicif and only if ( f ( X t )) t is a martingale. See [11, Chapter 5] for more on martingales.1.2. Statement of main results.
Our main result is a proof of Conjecture 1.1in the case that G is a virtually solvable group. We recall that any virtually solv-able group is in particular amenable, yet many amenable groups are not virtuallysolvable. Theorem 1.4.
Let G be a finitely generated virtually solvable group and let µ be acourteous probability measure on G . If dim HF k ( G , µ ) < ∞ for some k ≥ , then G is virtually nilpotent. Theorem 1.4 proves the implication (4) ⇒ (1) of Conjecture 1.1 assuming G is virtually solvable. All the other implications were previously known to holdwithout the assumption that G is virtually solvable: The implication (1) ⇒ (2) isa standard computation and follows from the Bass-Guivarc’h formula [3, 16]. Theimplication (2) ⇒ (3) is by Kleiner [20] via the method of Colding & Minicozzi[9], although strictly speaking, both Kleiner’s proof and the finitary version ofShalom and Tao [26] only deal with finitely supported measures. The implication(3) ⇒ (4) is trivial.A linear group is one which can be embedded in a some group of matrices overa field. Another direct but useful corollary of our result concerns linear groups: Corollary 1.5.
Let G be a finitely generated linear group and let µ be a courteousprobability measure on G . If there exists k ≥ such that dim HF k ( G , µ ) < ∞ then G is virtually nilpotent. ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 7
Proof.
Let ( G , µ ) be as above. By Proposition 1.3, G is amenable. By the Titsalternative [27], an amenable finitely generated linear group is virtually solvable.It follows by Theorem 1.4 that G is virtually nilpotent. ⊓⊔ Our proof of Theorem 1.4 is based on the following theorem, which is an explicitconstruction of a positive harmonic function of linear growth for finitely generatedsubgroups of the affine group. Let F be a field. The affine group of F is definedby(3) A ( F ) := (cid:8) x λx + c : λ ∈ F × , c ∈ F (cid:9) (here F × denotes the multiplicative group of invertible elements). Theorem 1.6.
Let G be a finitely generated subgroup of A ( F ) which is not virtuallyabelian. Suppose that µ is a courteous measure on G . Then, there exists a positive,non-constant, µ -harmonic function f : G → [0 , ∞ ) that has linear growth.Moreover, the vector space spanned by the orbit of f under the G -action isinfinite dimensional; i.e. dim span ( G f ) = ∞ . We prove Theorem 1.6 in Sections 4 and 5. Our construction of the function f is a generalization of a particular construction of a positive harmonic functionon the lamplighter group, which we now recall: Let G < (cid:2) F p ( x ) × F p ( x )0 1 (cid:3) be the lamplighter group with lamps in F p which can be defined by: G = (cid:8)(cid:2) x n c (cid:3) : c ∈ F p [ x, x − ] , n ∈ Z (cid:9) . The function f p is given by:(4) f p ( x ) = lim k →∞ r k P x [ c ( σ r k ) ∈ F p [ x ]] , where: • ( r k ) k is a certain increasing sequence of integers. • σ r = inf { t ≥ | λ ( t ) | 6∈ [ − r, r ] } , and λ ( t ) denotes the upper-left entryof the random 2 × X t (where X t is the random walk at time t ). TOM MEYEROVITCH AND ARIEL YADIN • c ( t ) denotes the upper-right entry of the random 2 × X t .Subsequently to writing a preliminarily version of our results, the constructionabove was exploited and generalized in a different direction by Tointon [28] tocharacterize groups with the property that the space of all harmonic functions isfinite dimensional. Also, after finishing this paper we observed that a somewhatrelated construction of positive harmonic functions on affine groups appears in[2, 6]. However, it is not immediately clear if the results of [2, 6] can be directlyapplied. One missing piece is an estimate for the polynomial growth rate of thosefunctions.In Section 3 we carry out a reduction of Theorem 1.4 to the case where G isa subgroup of A ( F ) as in the assumption of Theorem 1.6. For this reduction weinvoke a theorem of Groves [15] regarding finitely generated solvable groups, seeTheorem 3.6.2. Further research directions and open questions
Before going into the proofs, let us mention some further research directionsand some conclusions from this work.2.1.
Consequences of Conjecture 1.1.
Much is known about random walkson finitely generated group G of polynomial growth, see e.g. [1, 18]. For example,the random walk is diffusive ; that is, E [ | X t | ] ≍ Ct . Another example regards the entropy of the random walk: since a ball has polynomial growth, we have that H ( X t ) = O (log t ) (for more on entropy and random walks see [19] and referencestherein).Thus, the following are also consequences of Conjecture 1.1: Conjecture 2.1.
Let µ be a courteous probability measure on G such that dim HF ( G , µ ) < ∞ . Then: • E [ | X t | ] = O ( t ) where ( X t ) t is a µ -random walk on G . ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 9 • H ( X t ) = O (log t ) where H ( · ) is the Shannon entropy of a discrete randomvariable. Conjecture 1.1 implies that if one space HF k is finite dimensional, then all ofthem are: Conjecture 2.2.
Suppose that µ is a courteous probability measure on a finitelygenerated group G such that there exists k with dim HF k ( G , µ ) < ∞ .Then, for every k we have that dim HF k ( G , µ ) < ∞ . A well-known open question in the subject is whether the Liouville propertyis a group invariant. We must restrict ourselves to a certain class of measures:Indeed, for the lamplighter group on Z , L ( Z ) = ( Z / Z ) ≀ Z , it is quite simple toconstruct finitely-supported but non-symmetric measures and infinitely supportedsymmetric but non-smooth measures which are non-Liouville. However, any cour-teous measure on L ( Z ) is Liouville [19, Section 6.3]. (Actually [19] proves this forfinitely supported symmetric adapted measures. The proof for courteous measuresis along the same lines, and does not require any new ideas. One may also use acoupling argument in the spirit of [5].)The following conjecture has been unresolved for quite some time: Conjecture 2.3.
Let µ, ν be two courteous probability measures on a finitely gen-erated group G . Then ( G , µ ) is Liouville if and only if ( G , ν ) is Liouville. Regarding harmonic functions of polynomial growth, the analogous question is:
Conjecture 2.4.
Let µ, ν be two courteous probability measures on a finitely gen-erated group G . Then, for any k , dim HF k ( G , µ ) = dim HF k ( G , ν ) . This conjecture has been verified for the class of virtually solvable groups in [22],building on results in the current paper, or more generally under the assumption that HF k ( G , µ ) and dim HF k ( G , ν ) are both finite. Note that Conjecture 2.3 is justthe k = 0 case of Conjecture 2.4, because dim BHF ( G , µ ) is either 1 or ∞ .Progress toward proving Conjecture 1.1 will probably require an understandingof the kernel of the G -action on HF k ( G , µ ). For example, if this kernel is trivialfor some large enough k , the group is linear, and our results hold in this case. Question 2.5.
Suppose that µ is a courteous measure on a finitely generated group G . Describe the kernel of the G -action on HF k ( G , µ ) . Locally compact metric groups.
Throughout this paper we consideredfinitely generated countable groups. However, the definition of the spaces HF k ( G , µ )can be formulated for any measured group admitting a left-invariant metric, andit is natural to attempt to extend our results to the more general setting. Thereader may verify that the proof of Theorem 1.6 does not actually require G to befinitely generated, rather that it should be equipped with a left invariant metricfor which the conclusion of Lemma 5.4 holds. It seems plausible that the reductionfrom Theorem 1.6 to Theorem 1.4 should hold for groups admitting a left-invariantmetric in some greater generality. For instance, a topological version of the Titsalternative is known [8].Some care is required when the word metric for a finitely generated group isreplaced by an arbitrary left-invariant metric. To illustrate the point, consider thegroup G = Z , with respect to the invariant metric d ( n, m ) = p | n − m | . It is nolong true that homomorphisms are Lipschitz with respect to this metric.2.3. Lipschitz harmonic functions.
We believe the positive harmonic function f appearing in Theorem 1.6 is in fact Lipschitz. The probability estimates requiredto prove this seem to be more delicate then those appearing in Section 5. A proofthat f is Lipschitz would allows us to conclude that any finitely generated solvablegroup G such that the space of Lipschitz harmonic functions is finite dimensional is ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 11 virtually nilpotent, slightly improving Theorem 1.4. Note that Kleiner’s strategyto prove Gromov’s Theorem only requires that dim
LHF ( G , µ ) < ∞ . Conjecture 2.6.
Let µ be a courteous probability measure on G . If dim LHF ( G , µ ) < ∞ then dim HF ( G , µ ) < ∞ . Reduction from solvable to subgroups of A ( F )3.1. Random walks and finite index subgroups.
The nature of our resultforces us to pass to finite index subgroups in the course of the proof. In this sectionwe review the basic correspondence between harmonic functions on a group andon a finite index subgroup.In this section G is a general finitely generated group, ( X t ) t is a random walkon G with jump distribution µ , where µ is a courteous probability measure on G .For a subgroup H < G , define the hitting time τ H = inf { t ≥ X t ∈ H } . Wesay that H is a recurrent subgroup of G if τ H < ∞ a.s. It is well known that asubgroup of finite index is always recurrent. Furthermore, the expectation of τ H isequal to [ G : H ] (see e.g. [17] for a development of such relations in this context).A random variable X has an exponential tail if P [ | X | > t ] < Ce − ct for someconstants C, c >
0. It is straightforward to see that X has an exponential tail ifand only if E [ e α | X | ] < ∞ for some α > Lemma 3.1.
Let [ G : H ] < ∞ . For any adapted measure µ , τ H has an exponentialtail.Proof. Because µ is adapted, ( H X t ) t is an irreducible Markov chain on the finiteset H \ G . τ H is the first time this chain returns to the coset H . For any irreducibleMarkov chain on a finite set, hitting times always have an exponential tail. ⊓⊔ For a recurrent subgroup H the hitting measure is the a probability measureon H defined by µ H ( x ) = P [ X τ H = x ] . Note that a measure µ on a metric group G is smooth if and only if the lengthof a µ -random element of G has an exponential tail. Lemma 3.2.
Let µ be an adapted smooth measure on G and H ≤ G a subgroupof finite index. The hitting measure µ H is also a smooth measure.Proof. Let Z = Z H := S · S · · · S τ H , with S , S , . . . independently and identi-cally distributed according to µ . Note that for non-negative random variables, theproperty of having exponential tail is monotone with respect to stochastic domi-nation. Clearly, | Z | ≤ P τ H k =1 | S k | . However, τ H is not necessarily independent from( | S t | ) ∞ t =1 . We overcome this dependence as follows:Let x , . . . , x N ∈ G be a set of representatives for right H -cosets with x = 1 G ,so G = U Nj =1 H x j where N = [ G : H ]. Define a family of jointly independent G -valued random variables( S t ( i, j ) : t ∈ N , i, j ∈ { , . . . , N } )as follows: S t ( i, j ) is distributed according to µ conditioned on the event that S t ∈ x − i H x j (if this event has zero measure with respect to µ we make thearbitrary definition S t ( i, j ) = 1). Verify that Z is equal in distribution to arandom variable of the following form: τ H Y k =1 S k ( ξ k − , ξ k ) , Where ξ = 1 and ξ , ξ , ξ are a sequence of random variables taking values in { , . . . , N } , whose law is determined by the finite state Markov chain P [ ξ t +1 = j | ξ t = i ] = µ (cid:0) x − i H x j (cid:1) . ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 13
Since ( S t ) t all have an exponential tail, it follows that ( S t ( i, j )) t,i,j also have anexponential tail. Let ˆ S t := max i,j | S t ( i, j ) | . Then ˆ S t also has an exponential tail,and | Z | is stochastically dominated by P τ H t =1 ˆ S t . Now, note that τ H is equal indistribution to inf { t ≥ ξ t = 1 } , and also, ( ˆ S t ) t are independent of ( ξ t ) t . Since τ H , ˆ S , ˆ S , . . . have exponential tails and are jointly independent, it follows that | Z | is stochastically dominated by W := τ H X t =1 ˆ S t , where ( ˆ S t ) t , τ H are all independent Z + -valued random variables, ( ˆ S t ) t are i.i.d andall have an exponential tail.We are left with showing that there exists α > E [ e αW ] < ∞ .We know that since ( ˆ S t ) t all have an exponential tail, there exists β > E [ e β ˆ S t ] < ∞ . Similarly for τ H , there exists γ > E [ e γτ H ] < ∞ .Dominated convergence guaranties that E [ e β ˆ S t ] is continuous in β and E [ e β ˆ S t ] → β →
0. Thus, we may choose α > E [ e α ˆ S t ] < e γ . Withthis choice we have by independence of τ H , ( ˆ S t ) t , E [ e αW ] = ∞ X k =0 P [ τ H = k ] · k Y t =1 E [ e α ˆ S t ] ≤ ∞ X k =0 P [ τ H = k ] · e γk = E [ e γτ H ] < ∞ . ⊓⊔ Another property we wish to explore is the relation between HF k ( G , µ ) and HF k ( H , µ H ). Lemma 3.3.
Let µ be a courteous probability measure on G . Let ( X t ) t be a µ -random walk on G . Then, for any k there exists a constant C = C k such that forevery t ≥ , (5) E [ | X t | k ] ≤ C k · t k . Consequently, for any x ∈ G , (6) E x [ | X t | k ] ≤ C k · ( t + | x | ) k . Proof.
By the triangle inequality we have | X t | ≤ | X | + P tj =1 | S j | , where S j = X − j − X j is the jump at time j . So(7) E [ | X t | k ] ≤ E t X j =1 | S j | ! k = X ~j ∈{ ,...,t } k E [ k Y i =1 | S j i | ] . Let C k = max ≤ n ≤ k ( E [ | S | n ]) k/n . It follows that(8) E [ k Y i =1 | S j i | ] ≤ C k for all ~j ∈ { , . . . , t } k . Thus (5) follows from (7) and (8).To prove (6), note that E x | X t | k = E | xX t | k ≤ E [( | x | + | X t | ) k ]= k X j =0 (cid:18) kj (cid:19) | x | j · E | X t | k − j ≤ k X j =0 (cid:18) kj (cid:19) | x | j · C k − j t k − j . Taking C ′ k = max j ≤ k C j we get that E x | X t | k ≤ C ′ k · ( t + | x | ) k . ⊓⊔ The following lemma shows that µ -harmonic functions on G correspond bijec-tively to µ H -harmonic functions on H : Proposition 3.4.
Let G be a finitely generated group, µ a courteous measureon G and H ≤ G a subgroup of finite index. For any k ≥ , the restriction ofany f ∈ HF k ( G , µ ) to H is µ H -harmonic and in HF k ( H , µ H ) . Conversely, any ˜ f ∈ HF k ( H , µ H ) is the restriction of a unique f ∈ HF k ( G , µ ) . Thus, the restrictionmap is a linear bijection from HF k ( G , µ ) to HF k ( H , µ H ) . ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 15
Proof.
Step I: Extension.
Let ˜ f ∈ HF k ( H , µ H ). Define f : G → C by f ( x ) := E x [ ˜ f ( X τ )]where τ = τ H is the return time to H and ( X t ) t is a µ -random walk on G .We now wish to show that f is well-defined (equivalently, the expectation E x [ | ˜ f ( X τ ) | ] is finite), and that f ∈ HF k ( G , µ ).Note that since [ G : H ] < ∞ and G is finitely generated so is H . Also, withresect to any choices of finite symmetric generating sets G = h S i and H = h ˜ S i the corresponding metrics are bi-Lipschitz. Namely, there exist C > x ∈ H we have that C − | x | S ≤ | x | ˜ S ≤ C · | x | S (see for instance Corollary24 on page 89 of [10]). Thus, since X τ ∈ H and ˜ f ∈ HF k ( H , µ H ), we have that | ˜ f ( X τ ) | ≤ C · | X τ | k for some constant C > x ∈ G , since { τ > t } = { τ ≤ t } c ∈ F t := σ ( X , . . . , X t ), and since S t +1 := X − t X t +1 is independent of F t , we get that | f ( x ) | ≤ C · E x [ | X τ | k ] = C | x | k + C · ∞ X t =0 E x [ { τ>t } · ( | X t +1 | k − | X t | k )] ≤ C | x | k + C · ∞ X t =0 E x [ { τ>t } · (( | X t | + | S t +1 | ) k − | X t | k )]= C | x | k + C · ∞ X t =0 k X j =1 (cid:18) kj (cid:19) E [ | S t +1 | j ] · E x [ { τ>t } | X t | k − j ] . Using Lemma 3.3 we have that there exists some constant C k > (cid:0) E x [ { τ>t } | X t | k − j ] (cid:1) ≤ P x [ τ > t ] · E x [ | X t | k − j ) ] ≤ P x [ τ > t ] · C k · ( | x | + t ) k − j ) . Since τ has an exponential tail, setting M k = max j ≤ k E [ | S t +1 | j ], we have | f ( x ) | ≤ C | x | k + C · C k · M k · ∞ X t =0 e − ct k X j =1 (cid:18) kj (cid:19) ( | x | + t ) k − j = C | x | k + C · C k · M k · ∞ X t =0 e − ct (( | x | + t + 1) k − ( | x | + t ) k ) = O ( | x | k ) . This proves that f is well-defined and that k f k k < ∞ . A straightforward exam-ination of the definitions reveals that f is then µ -harmonic on G , and f (cid:12)(cid:12) H ≡ ˜ f .Thus f ∈ HF k ( G , µ ).To recap: for every ˜ f ∈ HF k ( H , µ H ) the extension f ( x ) := E x [ ˜ f ( X τ )] is afunction f ∈ HF k ( G , µ ). Step II: Restriction.
To show that this is indeed a unique extension, it sufficesto show that if f (cid:12)(cid:12) H ≡ f ∈ HF k ( G , µ ) then f ≡ G . Indeed, if f ∈ HF k ( G , µ ) then ( f ( X t )) t is a martingale.Also, ( f ( X τ ∧ t )) t is uniformly integrable: E x [ | f ( X τ ∧ t ) | ] ≤ C · E x [ | X τ ∧ t | k ] = C · E x [ | X τ | k { τ ≤ t } ] + C · E x [ | X t | k { τ>t } ] ≤ C · E x [ | X τ | k ] + C · E x [ | X t | k { τ>t } ] . We have already seen above that E x [ | X τ | k ] ≤ C k | x | k for some C k >
0. Also,Lemma 3.3 guaranties that because τ has an exponential tail, (cid:0) E x [ | X t | k { τ>t } ] (cid:1) ≤ P x [ τ > t ] · E x [ | X t | k ] ≤ e − ct · C k · ( | x | + t ) k . Thus, we obtain thatsup t E x [ | f ( X τ ∧ t ) | ] ≤ C · C k · | x | k + C · C k · sup t e − ct ( | x | + t ) k = O ( | x | k ) . So ( f ( X τ ∧ t )) t is uniformly integrable indeed.Thus, we may apply the Optional Stopping Theorem [11, § E x [ f ( X τ )] = f ( x ). Hence, if f ( x ) = 0 for all x ∈ H , then f ( X τ ) = 0 and so f ≡ G .This shows that the linear map f f (cid:12)(cid:12) H is injective on HF k ( G , µ ). ⊓⊔ ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 17
The reduction.
We will now complete the proof of Theorem 1.4 assumingTheorem 1.6.The dimension of HF k can only decrease when passing to a quotient group:Whenever G /N is a quotient of G , we have dim HF k ( G /N, µ ◦ π − ) ≤ dim HF k ( G , µ )where π : G → G /N is the canonical projection. Also, by Proposition 3.4, re-stricting to a finite index subgroup does not change the dimension of HF k . ThusTheorem 1.4 follows from the following proposition. Proposition 3.5 (see [7]) . Let G be a finitely generated solvable group which is notvirtually nilpotent. Then there exists a quotient G /N of G which is not virtuallynilpotent, and has a finite index subgroup H < G /N which embeds in A ( F ) forsome field F . This precise reduction is carried out in [7]. We provide a few details. Theargument is based on the following theorem of Groves:
Theorem 3.6 (Groves [15]) . Let G be a finitely generated solvable group that isjust non virtually nilpotent (that is, for any non-trivial normal subgroup N ⊳ G wehave that
G/N is virtually nilpotent). Then there exists a finite index subgroup [ G : H ] < ∞ such that H is isomorphic to a subgroup of the affine group over afield K . A proof of Theorem 3.6 also appears in [7, Section 4].
Outline of proof of Proposition 3.5.
We assume that G is solvable but not virtu-ally nilpotent, and µ a courteous measure on G .Every finitely generated non virtually nilpotent group has a just non virtuallynilpotent quotient (see e.g. Claim 2 in the beginning of Section 5 of [7]).So let G /N be a just non virtually nilpotent quotient of G . Since G is finitelygenerated and solvable, so is G /N . By Theorem 3.6 G /N has a finite index subgroup H that is isomorphic to a subgroup of the affine group over some field F . ⊓⊔ We now complete the section with
Proof of Theorem 1.4 assuming Theorem 1.6.
Let G be a finitely generated virtu-ally solvable group, µ a courteous measure on G and assume that G is not virtuallynilpotent.By Proposition 3.5, there exists N ⊳ G and a finite index subgroup H < G /N such that G /N is not virtually nilpotent and H is isomorphic to a subgroup of A ( F )for some field F . Since H is finite index in a quotient of G , we have by Proposition3.4 that dim HF ( H , ν ) ≤ dim HF ( G , µ ), for some courteous measure ν (obtainedby projecting µ from G to G /N and then taking the induced hitting measureon H ). Because G /N is not virtually nilpotent, it cannot be that H is virtuallyabelian. Theorem 1.6 now tells us that dim HF ( G , µ ) ≥ dim HF ( H , ν ) = ∞ . ⊓⊔ Random walks on the reals
In this section we collect probability estimates and results regarding randomwalks on the real line. These estimates will serve us in the following section toprove Theorem 1.6. As this is not the main focus of this paper, we omit proofsfor some standard statements.In the following, ( Y t ) t is a sequence of real valued random variables such that Z t := ( Y t − Y t − ) t are i.i.d. symmetric random variables of mean 0. Thus, ( Y t ) t isa martingale. We assume that the random variables Z t have an exponential tail;that is, there exists ε > E [ e ε | Z t | ] < ∞ .For any set A ⊂ R , τ A is the hitting time of A and σ A the exit time of A , i.e. τ A = inf { t ≥ Y t ∈ A } and σ A = τ R \ A . ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 19
For r > τ r = τ [ − r,r ] and σ r = σ [ − r,r ] . P y , E y denote the probability measureand expectation conditioned on Y = y .4.1. Standard lemmas.
The following three lemmas are relatively standard, andwe omit the proofs.
Lemma 4.1.
There exist constants c, C > (depending only on the distributionof Z ) such that for any r > and any | y | ≤ r , E y [ σ r ] ≤ Cr and P y [ σ r > t ] ≤ e − ct/r . Lemma 4.2.
There exist constants
C, ε > such that for any r > and | y | < r ,for all z > , P y [ ∃ t ≤ σ r : | Z t | > z ] ≤ Cr e − εz + e − r . We also need an estimate for the probability that ( Y t ) t exits an interval fromthe left (or from the right). Lemma 4.3.
There exists a universal constant δ > such that for all y ∈ ( − r, r ) ,as r → ∞ , P y [ τ ( r, ∞ ) < τ ( −∞ , − r ) ] = y + r r · (1 + O ( r − δ )) . For random walks with bounded jumps, the proof is quite easy and standard.Without the bounded jumps assumption, there are some technical details to dealwith. This can be handled for instance by bounding the probability of a very largejump occurring before exiting the interval, using Lemma 4.2. Again, we omit theproof, pointing the reader for instance to [21] for some results and proofs of similarflavor.4.2.
Specific lemmas.
The next lemma is somewhat specifically tailored for ourapplication. It asserts that the random walk is very unlikely to spend too muchtime near the endpoints of an interval before exiting, even if we condition on the side from which the random walk exists. We therefore include a full proof. Asmentioned, our bounds are not optimal, and we focus on brevity.
Lemma 4.4.
Let V m be the time spent by ( Y t ) t in the interval [0 , m ] until exiting [0 , r ] ; that is, V m = X t { Y t ∈ [0 ,m ] ,σ [0 ,r ] >t } = σ [0 ,r ] − X t =0 { Y t ∈ [0 ,m ] } . Let B = (cid:8) τ ( r, ∞ ) < τ ( −∞ , (cid:9) . There exists a constant c > (depending only on thedistribution of Z ) such that for all < m < r and < y < r the following holds: P y [ V m > v ] ≤ e − cv/m and P y [ V m > v , B ] ≤ e − cv/m · sup x ∈ [0 ,m ] P x [ B ] . Proof.
Using Lemma 4.1 choose
C > m ) be such that P [ σ m ≤ Cm ] > . For every y ∈ [0 , m ], after translating by y and using the fact that thewalk is symmetric, P y [ τ ( −∞ , ≤ Cm ] = P [ τ ( −∞ , − y ) ≤ Cm ] ≥ P [ τ ( −∞ , − y ) < τ ( y, ∞ ) , σ y ≤ Cm ]= P [ σ y ≤ Cm ] ≥ P [ σ m ≤ Cm ] ≥ , the last line following since σ y ≤ σ m a.s. Since this is uniform over all y ∈ [0 , m ]we have that for any 0 < y < r , taking s := ⌈ Cm ⌉ , P y [ V m > v ] ≤ sup z ∈ [0 ,m ] P z [ τ ( −∞ , > s ] · sup x> P x [ V m > v − s ] ≤ · sup x> P x [ V m > v − s ] ≤ · · · ≤ (cid:0) (cid:1) ⌊ v/s ⌋ . The proof of the second assertion is similar. P y [ V m > v , B ] ≤ · sup x> P x [ V m > v − s , B ] ≤ · · · ≤ (cid:0) (cid:1) ⌊ v/s ⌋− · sup x> P x [ V m ≥ , B ] . The strong Markov property at time τ [0 ,m ] gives thatsup x> P x [ V m ≥ , B ] ≤ sup x ∈ [0 ,m ] P x [ B ] . ⊓⊔ ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 21
For the next lemma we need the notion of a maximal separated subset. Givenan interval I ⊂ R and r > MS r ( I ) = MS r (( Y t ) t , I ) denote the maximalcardinality of a 1-separated subset of I ∩ { Y , Y , . . . , Y σ r − } ; that is MS r ( I ) = max (cid:8) | A | : A ⊂ I ∩{ Y , Y , . . . , Y σ r − } and ∀ a = a ′ ∈ A, | a − a ′ | ≥ (cid:9) . Lemma 4.5.
For any q > there exists a constant C > such that for all y ≤ with r > {− y, q } , and any n ≤ √ r , P y [ MS r (( −∞ , − q )) ≤ n ] < C ( n + 1) r . Proof.
Define inductively: T = 0 and m = Y . For k > T k := inf { t > T k − : Y t ≤ m k − − } and m k := Y T k . So ( T k ) k are the successive times the random walk ( Y t ) t passesbelow its minimum by at least 1. Note that by definition { m , m , . . . , m k } isa 1 separated subset of ( −∞ , m k ≤ m k − − ℓ = ⌈ q ⌉ + 1, the set { m ℓ , . . . , m ℓ + n } is a 1-separated subset of ( −∞ , − q ) of size n + 1. Thus, the event { T ℓ + n < σ r } implies the event {MS r (( −∞ , − q )) > n } .Now, let E = {∀ t ≤ σ r , | Z t | ≤ r/ n } . By Lemma 4.2, P y [ E c ] ≤ Cr e − εr/ n forsome constants C, ε that depend only on the distribution of Z t . By adjusting theconstant in the statement of the lemma we may assume without loss of generalitythat ℓ < n . Thus, ( ℓ + n ) · r n < r . Hence the event { T ℓ + n ≥ σ r } ∩ E implies theevent (cid:8) T ℓ + n ≥ σ r = τ ( r, ∞ ) (cid:9) . Since T < σ r a.s., the event { T ℓ + n ≥ σ r } ∩ E impliesthat there exists 0 < k ≤ ℓ + n such that T k − < σ r and T k ≥ σ r = τ ( r, ∞ ) . Theprobability of this can be bounded by the strong Markov property at time T k − and Lemma 4.3, P y [ T k − < σ r , T k ≥ σ r = τ ( r, ∞ ) ] ≤ sup y ∈ [ − r, − ( k − P y [ T ≥ σ r = τ ( r, ∞ ) ] ≤ sup y ∈ [ − r, − ( k − P y [ τ ( −∞ ,y − > τ ( r, ∞ ) ] ≤ Cr .
Thus, P y [ MS r (( −∞ , − q )) ≤ n ] ≤ P y [ E c ] + ℓ + n X k =1 P y [ T k − < σ r , T k ≥ σ r = τ ( r, ∞ ) ] ≤ Cr e − εr/ n + C ( ℓ + n ) r . The lemma follows since for n ≤ √ r we have that r e − εr/ n ≤ C ′ r − for someconstant C ′ > ⊓⊔ A positive harmonic function for subgroups of A ( F )In this section we prove Theorem 1.6, using the random-walk estimates fromthe previous section.We make the following obvious identification: A ( F ) := (cid:8)(cid:2) λ c (cid:3) : c ∈ F , λ ∈ F × (cid:9) . For x = (cid:2) λ c (cid:3) ∈ A ( F ), we denote:(9) c ( x ) = c, λ ( x ) = λ Let G < A ( F ) be as in the statement of Theorem 1.6. Some reductions will beuseful. Lemma 5.1.
Let F be a field and suppose G is a finitely generated subgroup of A ( F ) . If the set λ ( G ) = { λ : (cid:2) λ c (cid:3) ∈ G } is contained in the group of roots ofunity of F × then G is virtually abelian.Proof. If λ ( G ) is contained in the group of roots of unity of F × then it is a finitegroup, and the kernel of the homomorphism λ : G → F × is an abelian group,consisting of elements of the form (cid:2) c (cid:3) . ⊓⊔ The following simple version of Kronecker’s Theorem is used in the proof of theTit’s Alternative [27]. See for instance [8, Section 2] for a generalization and briefdiscussion.
ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 23
Lemma 5.2.
Let F be a finitely generated field and let α ∈ F be an element whichis not a root of unity. Then there exists a local field K with absolute value | · | andan embedding of fields ι : F → K , such that | ι ( α ) | > . In our case, since G is a finitely generated subgroup of the affine group over F , wecan assume without loss of generality that F is a field generated by { λ, c : (cid:2) λ c (cid:3) ∈ S } , where S is a finite generating set for G . Keep in mind that F can be ofpositive or zero characteristic. Thus, there is an embedding ι : F → K , where K is a local field such that | ι ( λ ) | > (cid:2) λ c (cid:3) ∈ G , and | · | is the absolutevalue on K . Since ι is an embedding of fields, it induces an embedding of groups ι : A ( F ) → A ( K ).With the above considerations in mind, from now on we assume F is a local fieldwith absolute value | · | ; the group G ≤ A ( F ) is a finitely generated, non-abeliancountable group such that | λ | > (cid:2) λ c (cid:3) ∈ G ; and µ is a courteousprobability measure on G . Furthermore, we assume without loss of generalitythat there is an element x ∈ G of the form x = (cid:2) λ
00 1 (cid:3) with | λ | >
1. Indeed, If (cid:2) λ c (cid:3) ∈ G with | λ | >
1, we get an element of the correct form by conjugating G with (cid:2) − λ ) − c (cid:3) . Lemma 5.3.
There exist a finite index subgroup H < G and an element z ∈ H ofthe form (10) z = (cid:2) c (cid:3) c = 0 such that the hitting measure satisfies µ H ( z ) > Proof.
Because G is not abelian, it contains a non-trivial commutator. Further-more, since µ is adapted, we can find non-commuting elements a, b ∈ G so that µ ( a ) > µ ( b ) >
0. Because a and b are non-commuting, it follows that a = 1 and b = 1 so we can assume that λ ( a ) = 1 and λ ( b ) = 1, otherwise we canconclude the proof with H = G and z = a or z = b . By possibly replacing a with a − we can further assume that λ ( ab ) = 1. Let z := [ a, b ]. It follows that indeed z = (cid:2) c (cid:3) with c = 0. Because λ ( G ) ⊂ F × is a finitely generated abelian groupit is residually finite, so there exists a finite index subgroup Λ < λ ( G ) so that λ ( a ) , λ ( b ) , λ ( ab ) Λ . Let H = λ − (Λ ), then H < G is a finite index subgroupand z ∈ H . By the construction of H , we have a, ab, aba − H . It follows that µ H ( z ) ≥ µ ( a − ) µ ( b − ) µ ( a ) µ ( b ) > ⊓⊔ Thus, by replacing ( G , µ ) with ( H , µ H ) we further assume without loss of gen-erality that µ ( z ) > z ∈ G satisfying (10).We introduce a bit more notation. Let(11) ρ ( x ) = − log | λ ( x ) | . ( X t ) t denotes a discrete time random walk on G such that the jumps S t = X − t − X t are identically distributed with distribution µ . P x , E x denote the prob-ability measure and expectation of random walks with X = x . Specifically, P x [ X t +1 = y | X t = z ] = µ ( z − y ).As in Section 4, for A ⊂ R , we use the notation τ A = inf { t ≥ ρ ( X t ) ∈ A } and σ A = inf { t ≥ ρ ( X t ) A } . For r > τ r = τ [ − r,r ] , σ r = σ [ − r,r ] .We define a function f : G → [0 , ∞ ) as follows:(12) f ( x ) = lim k →∞ r k P x [ | c ( X σ rk ) | < , where ( r k ) k is some increasing sequence of integers for which the limit in (12)exists. We will prove Theorem 1.6 by showing that f given by (12) satisfies allthe requirements.5.1. Preliminary lemmas.
There are fairly straightforward bounds on | ρ ( x ) | and | c ( x ) | which we now note, omitting the simple proof. ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 25
Lemma 5.4.
There exists a constant
K > so that for any x ∈ G | ρ ( x ) | ≤ K | x | and | c ( x ) | ≤ e K | x | The following lemma is crucial for proving that the function constructed has atmost linear growth, and for proving it is non-constant.
Lemma 5.5.
There exists a constant
K > (depending only on G and µ ) suchthat for all x ∈ G and r > with < ρ ( x ) < r , P x [ | c ( X σ r ) − c ( X ) | > , τ ( r, ∞ ) < τ ( −∞ , ] ≤ Kr .
Proof.
By Lemma 5.4, because µ is smooth, there exists ε > E [ | c ( S ) | ε ] = E [ | c ( X ) | ε ] < ∞ . Let B = (cid:8) τ ( r, ∞ ) < τ ( −∞ , (cid:9) be the event that the walk ( ρ ( X t )) t exits [0 , r ] from the right.Define A m = σ [0 ,r ] − X t =0 | c ( S t +1 ) | { ρ ( X t ) ∈ [ m,m +1) } . The quantity A m will be used to control the contribution to | c ( X σ r ) − c ( X ) | fromincrements of c at the times ρ ( X t ) is in the interval [ m, m + 1), before exiting [0 , r ].We have that for any x with ρ ( x ) >
0, by Lemma 4.3 and the Markov propertyat time t , E x [ | c ( S t +1 ) | ε { t<σ r } { ρ ( X t ) ∈ [ m,m +1) } B ] ≤ X s µ ( s ) | c ( s ) | ε · sup y : ρ ( y )
0, by Markov’s inequality, P x [ | c ( S t +1 ) | > β , t < σ r , ρ ( X t ) ∈ [ m, m + 1) , B ] = P x [ | c ( S t +1 ) | ǫ > β ǫ , t < σ r , ρ ( X t ) ∈ [ m, m + 1) , B ] ≤≤ Cmβ ǫ r · P x [ ρ ( X t ) ∈ [ m, m + 1) , t < σ [0 ,r ] ](13)Define V m = σ [0 ,r ] − X t =0 { ρ ( X t ) ∈ [ m,m +1) } , the number of visits to [ m, m +1) by the walk ( ρ ( X t )) t , before exiting [0 , r ]. Lemma4.4 together with Lemma 4.3 tell us that for some constant c > P x [ V m > v ] ≤ e − cv/m and P x [ V m > v , B ] ≤ e − cv/m · Cmr .
So we may choose
C > m , P x [ V m > Cm , B ] ≤ r · e − m and E x [ V m ] ≤ Cm . Summing (13) over t , we have that P x [ ∃ t < σ r , | c ( S t +1 ) | > β , ρ ( X t ) ∈ [ m, m + 1) , B ] ≤ E x [ V m ] · Cmβ ε r ≤ C ′ m β ε r . Taking β = C − m − ( e/ m , P x [ A m e − m > − m , B ] ≤ P x [ A m e − m > − m , V m ≤ Cm , B ] + P x [ V m > Cm , B ] ≤ C ′ m β ε r + 1 r e − m ≤ r e − δm , ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 27 for some constant δ = δ ( ε ) >
0. Summing over m ≥ P x [ ∃ m ≥ , A m e − m > − m , B ] ≤ Cr .
Since c ( xy ) = c ( x ) + λ ( x ) c ( y ), we have c ( X σ r ) − c ( X ) = σ r − X t =0 c ( S t +1 ) · λ ( X t ) , and so | c ( X σ r ) − c ( X ) | ≤ σ r − X t =0 | c ( S t +1 ) | · e − ρ ( X t ) ≤ X m A m e − m . On the event B we have that A m = 0 for all m <
0. So the event B ∩{∀ m ≥ , A m e − m ≤ − m } implies that | c ( X σ r ) − c ( X ) | ≤ X m ≥ − m = 2 . Thus, for any x ∈ G with ρ ( x ) > P x [ | c ( X σ r ) − c ( X ) | > , B ] ≤ Cr . ⊓⊔ As in the previous section, just before Lemma 4.5, we denote the maximal cardi-nality of a 1-separated subset of { ρ ( X t ) : t < σ r }∩ I by MS r ( I ) = MS r (( ρ ( X t )) t , I ).The following is a key lemma for proving the function f is non-constant. Lemma 5.6.
There exist
C, q, ǫ > , depending only on G , µ , such that P x [ | c ( X σ r ) | < | MS r (( −∞ , − q )) = n ] ≤ Ce − ǫn . for all n ∈ Z + , r > and x ∈ G .Proof. By adjusting the constant in the statement of the lemma we may assumewithout loss of generality that n ≥
1. Let z ∈ G satisfy (10) with µ ( z ) >
0. Suchan element exists by our assumptions. Choose q > q > log (cid:18) | c ( z ) | (1 − P ∞ k =1 e − k ) (cid:19) , and let I = ( −∞ , − q ).Consider the set R = { ρ ( X t ) : t < σ r } ∩ I . Define the event(15) E n = {MS r ( I ) = n } Note that the event E n is measurable with respect to the (set-valued) randomvariable R .Assume we are in the event E n . Let A = A ( R ) ⊂ I ∩ R be a 1-separated setof size n . Formally, A is a set-valued random variable which is measurable withrespect to the random variable R , and which on the event E n is a.s. a 1-separatedsubset A ⊂ I ∩ R of size | A | = n .For ρ ∈ R let t ρ = inf { t ≥ ρ ( X t ) = ρ } and T = { t ρ : ρ ∈ A } . For t ≥ ξ t = S t +1 = z − S t +1 = z − T ′ = { t ∈ T : | ξ t | = 1 } . Note that S t +1 = z ξ t for t ∈ T ′ .Let ( ˇ S t ) t denote the sequence obtained from ( S t ) t by changing every occurrenceof z − at times in T ′ to z : ˇ S t = z t ∈ T ′ S t otherwiseWe claim that:Conditioned on the event E n and on R , the random variables( | ξ t ρ | ) ρ ∈ A are i.i.d. and P [ t ρ ∈ T ′ | ρ ∈ A , R ] = 2 µ ( z ).(16)For now let us proceed with the proof assuming (16). LetΠ = c ( X ) + σ r − X t =0 { t T ′ } λ ( X t ) c ( ˇ S t +1 ) . ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 29
Since λ ( z ) = 1, it follows that λ ( ˇ S j ) = λ ( S j ) so λ ( X t ) = Q t − j =1 λ ( ˇ S j ) is measurablewith respect to ( ˇ S t ) t , and hence also Π is measurable with respect to ( ˇ S t ) t .Because c ( X σ r ) = c ( X ) + P σ r − t =0 λ ( X t ) c ( S t +1 ) we have: c ( X σ r ) = Π + X t ∈ T ′ λ ( X t ) c ( S t +1 ) = Π + c ( z ) X t ∈ T ′ λ ( X t ) ξ t Note that for any t ∈ T ′ we have that ρ ( X t ) ∈ R ⊂ I so − ρ ( X t ) > q .Also, for all ρ = ρ ′ ∈ A we have that | ρ − ρ ′ | ≥
1, which implies that for any ξ ′′ = ξ ′ ∈ {− , } T ′ , there is some ρ ∈ A such that (cid:12)(cid:12) c ( z ) X t ∈ T ′ λ ( X t )( ξ ′′ t − ξ ′ t ) (cid:12)(cid:12) ≥ | c ( z ) | e − ρ · − ∞ X k =1 e − k ! > , where the last inequality holds by (14) because e − ρ > e q . Thus, for any α ∈ F there is at most one possible vector ξ ∈ {− , } T ′ for which (cid:12)(cid:12) α + c ( z ) · X t ∈ T ′ ξ t λ ( X t ) (cid:12)(cid:12) < . Note that the set T ′ is measurable with respect to ( ˇ S t ) t . From the fact that( S t ) t are i.i.d and the definition of ( ˇ S t ) t , it follows that conditioned on ( ˇ S t ) t thedistribution of ( ξ t ) t ∈ T ′ is uniform on {− , } T ′ . Thus we have P x [ | c ( X σ r ) | < |E n ] = E x P x h(cid:12)(cid:12) Π + c ( z ) · X t ∈ T ′ ξ t λ ( X t ) (cid:12)(cid:12) < (cid:12)(cid:12)(cid:12) ( ˇ S t ) t , E n i ≤ E x [2 −| T ′ | |E n ] ≤ P x [ | T ′ | < µ ( z ) n |E n ] + 2 − µ ( z ) n . (17)By (16), conditioned on the event E n and on the set R , the distribution of | T ′ | isbinomial-( n, µ ( z )). Using a well known large deviation estimate for the binomialdistribution P x [ | T ′ | < µ ( z ) n |E n ] ≤ e − ǫn . for some constant ǫ = ǫ ( µ ( z )) >
0. So combined with (17) we have that P x [ | c ( X σ r ) | < |E n ] ≤ e − εn + 2 − µ ( z ) n . (18)We now justify (16) by providing an alternative description of the process ( S t ) t .The idea is that the process can be constructed by sampling independent stepswhich are conditioned not to be z or z − and then “spacing” them with a geomet-rically distributed number of steps each of which is equal to z or z − independentlywith equal probability. Formally:Start with a sequence ( ˆ S t ) t of i.i.d elements in G each distributed according to µ , conditioned on the event { ˆ S t
6∈ { z, z − }} , and another sequence ( Z t ) t of i.i.delements in G each distributed so that P [ Z t = z ] = P [ Z t = z − ] = . (19)Let ( T j ) ∞ j =1 be i.i.d. integer valued random variables with distribution P [ T j = k ] = (2 p ) k (1 − p ) for k = 0 , , , . . . , where p = µ ( z ). The processes ( T j ) ∞ j =1 , ( ˆ S t ) t and ( Z t ) t are jointly independent.Now obtain a new process ( ˜ S t ) t inductively as follows: Start with L [1] = T , j [1] = 1 and k [1] = 1 and define inductively( L [ t + 1] , j [ t + 1] , k [ t + 1]) = ( L [ t ] − , j [ t ] , k [ t ] + 1) L [ t ] > T j [ t ]+1 , j [ t ] + 1 , k [ t ]) otherwiseFor t = 1 , , . . . let ˜ S t = Z k [ t ] L [ t ] > S j [ t ] otherwiseIt follows that ( ˜ S t ) t are i.i.d, each distributed according to µ . Thus, we mayreplace ( S t ) t with ( ˜ S t ) t . ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 31
Note that the sets R = { ρ ( X t ) : t < σ r } ∩ I and A = A ( R ) are measurablewith respect to ( ˆ S j ) j . For any ρ ∈ T we have˜ S t ρ = Z k [ t ρ ] T j [ t ρ ] > S j [ t ρ ] otherwiseSo ˜ S t ρ ∈ { z, z − } if and only if T j [ t ρ ] >
0. Check that the process ( ˆ S j ) j is indepen-dent from T j [ t ρ ] and Z k [ t ρ ] . From the above and the independence of ( T j ) j , ( ˆ S j ) j and ( Z t ) t and the fact that P [ T j >
0] = 2 p , (16) now follows. ⊓⊔ Well defined and harmonic.Proposition 5.7.
There exists a constant
K > such that for all x ∈ G and r > such that − r < ρ ( x ) ≤ , P x [ | c ( X σ r ) | < ≤ Kr .
Proof.
Let q > I = ( −∞ , − q ) and MS r ( I ) be as in Lemma 5.6. P x [ | c ( X σ r ) | < ≤ ∞ X n =0 P x [ | c ( X σ r ) | < | MS r ( I ) = n ] · P x [ MS r ( I ) ≤ n ]By Lemma 4.5, for all n ≤ √ r we have P x [ MS r ( I ) ≤ n ] ≤ C ( n +1) r . So usingLemma 5.6, P x [ | c ( X σ r ) | < ≤ X n ≤√ r C ( n + 1) r e − ǫn + X n> √ r e − εn ≤ Kr . ⊓⊔ Proposition 5.8.
There exists a constant
C > , depending on G and µ , suchthat for all x ∈ G and r > | ρ ( x ) | , P x [ | c ( X σ r ) | < ≤ C max { ρ ( x ) , } r . Proof.
Let E = {∃ t < σ r , ρ ( X t ) ∈ ( − r/ , } and E = (cid:8) τ ( −∞ , < τ ( r, ∞ ) (cid:9) . We have:(20) P x [ | c ( X σ r ) | < ≤ P x [ E ∩ E c ] + P x [ | c ( X σ r ) | < , E ] + P x [ E c ]Note that because ρ ( X ) > − r the event E ∩ E c implies that the random walk( ρ ( X t )) t jumps across the interval ( − r , P x [ E ∩ E c ] ≤ P x (cid:2) ∃ t ≤ σ r : | ρ ( S t ) | > r (cid:3) ≤ Ce − εr , (21)for some constants C, ε .By the strong Markov property and Proposition 5.7, there exists
C > P x [ | c ( X σ r ) | < , E ] ≤ sup − r/ <ρ ( y ) < P y [ | c ( X σ r ) | < ≤ Cr .
By Lemma 4.3 there exists
C > P x [ E c ] ≤ ρ ( x ) + Cr .
The proof follows by applying the bounds in (21), (22) and (23) on the righthand side of (20). ⊓⊔ Proposition 5.9.
There exists an increasing sequence of integers ( r k ) k for whichthe limit in (12) exists for all x ∈ G . For such a sequence the function f definedby the limit in (12) satisfies f ∈ HF ( G , µ ) . That is f is µ -harmonic and thereexists some constant C > so that | f ( x ) | ≤ C ( | x | + 1) for all x ∈ G .Proof. For r ∈ N let f r : G → R + be given by f r ( x ) = r · P x [ | c ( X σ r ) | < · { r> | ρ ( x ) |} . ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 33
By Proposition 5.8, sup r f r ( x ) < ∞ for all x ∈ G , so by Arzel`a-Ascoli thereexists a subsequence ( r k ) along which there is pointwise convergence. Let f bethis subsequential limit.To see that f is µ -harmonic, note for r > | ρ ( x ) | we have P x [ | c ( X σ r ) | <
3] = X s µ ( s ) P xs [ | c ( X σ r ) | < . Thus f ( x ) = lim k →∞ r k · P x [ | c ( X σ rk ) | <
3] = X s µ ( s ) lim k →∞ r k · P xs [ | c ( X σ rk ) | <
3] = X s µ ( s ) f ( xs ) . It remains to observe that by Proposition 5.8 and Lemma 5.4, P x [ | c ( X σ r ) | < ≤ C max { ρ ( x ) , } r ≤ C ′ ( | x | + 1) r . Multiplying by r and taking limits this proves that f ( x ) ≤ C ′ ( | x | + 1). ⊓⊔ Non-constant.Proposition 5.10.
There exist constants
C, ǫ > depending only on G and µ ,such that for any x ∈ G and r > with | c ( x ) | < and C < ρ ( x ) < r , P x [ | c ( X σ r ) | < ≥ ǫρ ( x ) − Cr .
Proof. If | c ( x ) | < P x [ | c ( X σ r ) | < ≥ P x [ | c ( X σ r ) − c ( x ) | ≤ . Let B = (cid:8) τ ( r, ∞ ) < τ ( −∞ , (cid:9) . We have P x [ | c ( X σ r ) − c ( x ) | ≤ ≥ P x [ | c ( X σ r ) − c ( x ) | ≤ , B ] . By Lemmas 4.3 and 5.5, P x [ | c ( X σ r ) − c ( x ) | ≤ , B ] ≥ P x [ B ] − Cr ≥ ǫρ ( x ) − Cr . ⊓⊔ Proposition 5.11.
The function f defined in (12) is non-constant.Proof. By one of our assumptions, there exists x = (cid:2) λ
00 1 (cid:3) ∈ G with | λ | >
1, so x − n = (cid:2) λ − n
00 1 (cid:3) . By Proposition 5.10,lim n →∞ f ( x − n ) = + ∞ , so the function f is unbounded and in particular non-constant. ⊓⊔ Infinite dimensional orbit.Proposition 5.12.
There exists a constant
C > such that for all x ∈ G with | c ( x ) | > , and all r > | ρ ( x ) | , P x [ | c ( X σ r ) | < ≤ Cr .
Proof.
Let B = (cid:8) τ ( r, ∞ ) < τ ( −∞ , (cid:9) . Let E = {∃ t < σ r : ρ ( X t ) ∈ ( − r/ , } . Bythe strong Markov property P x [ | c ( X σ r ) | < , B c ] ≤ P x [ E c , B c ] + sup y : ρ ( y ) ∈ ( − r/ , P y [ | c ( X σ r ) | < Cr . UsingLemma 4.2 (as in the proof of Proposition 5.8), there are constants C, ε > P x [ E c , B c ] ≤ Ce − εr .If ρ ( x ) ≤ P x [ B ] = 0, so P x [ | c ( X σ r ) | < , B ] = 0. If ρ ( x ) > | c ( x ) | > C > P x [ | c ( X σ r ) | < , B ] ≤ P x [ | c ( X σ r ) − c ( x ) | > , B ] ≤ Cr .
Altogether, for any x ∈ G with | c ( x ) | > P x [ | c ( X σ r ) | <
3] = P x [ | c ( X σ r ) | < , B ] + P x [ | c ( X σ r ) | < , B c ] ≤ Cr . for some constant
C > ⊓⊔ ARMONIC FUNCTIONS OF LINEAR GROWTH ON SOLVABLE GROUPS 35
Proposition 5.13.
Let f : G → [0 , ∞ ) be the function given in (12) . Thereexist ( y n ) n ⊂ G such that the family ( f n := y n f ) n are infinitely many linearlyindependent functions.Specifically, dim span ( G f ) = ∞ .Proof. By our assumptions on G , we have elements z = (cid:2) c (cid:3) ∈ G with c = 0, and x = (cid:2) λ
00 1 (cid:3) ∈ G with | λ | >
1. Choose N large enough so that | λ N | · ( | λ N | − | ) · | c | > . Let y n = x Nn zx − Nn = (cid:2) λ Nn c (cid:3) and f n = y n f. A simple calculation shows that for m, n, j ∈ N : f n ( y m x − j ) = f ( y − n y m x − j ) = f (cid:0)(cid:2) λ − j ( λ Nm − λ Nn ) c (cid:3)(cid:1) If 1 ≤ n < m then (cid:12)(cid:12) ( λ Nm − λ Nn ) c (cid:12)(cid:12) ≥ | λ Nn | · (cid:12)(cid:12) | λ N ( m − n ) | − (cid:12)(cid:12) · | c | > . Using the symmetry between n, m , by Proposition 5.12, there exists a constant
C > n = m , we have | f n ( y m x − j ) | ≤ C for any j ∈ N . On theother hand, by Proposition 5.10 we havelim j →∞ f n ( y n x − j ) = + ∞ . It follows that for if α , . . . , α m ∈ C and α m = 0 then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X n =1 α n f n ( y m x − j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ | α m | · | f m ( y m x − j ) | − m − X n =1 | α n | · | f n ( y m x − j ) |≥ | α m | · f m ( y m x − j ) − m − X n =1 | α n | · C → ∞ as j → ∞ . We conclude that the functions ( f n ) n are indeed linearly independent. ⊓⊔ Proof of Theorem 1.6.
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Department of Mathematics, Ben Gurion University of the Negev, Be’er ShevaISRAEL.
E-mail address : { mtom, yadina }}