The group of quasi-isometries of the real line cannot act effectively on the line

We prove that the group $\mathrm{QI}^{+}(\mathbb{R})$ of orientation-preserving quasi-isometries of the real line is a left-orderable, non-simple group, which cannot act effectively on the real line $\mathbb{R}.$

Theorem 1.3 The quasi-isometry group QI C .R/ cannot act effectively on the real line R.
Other (uncountable) left-orderable groups that cannot act on the line are been known.For example, the germ group G 1 .R/, due to Mann [4] and Rivas; and the compact supported diffeomorphism group Diff c .R n / for n > 1, due to Chen and Mann [1]. 2 The group structure of QI.R/ Let QI.R C / (resp.QI.R /) be the quasi-isometry group of the ray OE0; C1/ (resp. .1; 0), viewed as subgroup of QI.R/ fixing the negative (resp.positive) part.
Lemma 2.1 QI.R/ D .QI.R C / QI.R // Ì ht i, where t 2 QI.R/ is the reflection t .x/D x for any x 2 R.
Proof Sankaran [9] proves that the group PL ı .R/ consisting of piecewise linear homeomorphisms with bounded slopes has a full image in QI.R/.Since every homeomorphism f 2 PL ı .R/ is of bounded distance to the map f f .0/ 2 PL ı .R/, we see that the subgroup PL ı;0 .R/ D ff 2 PL ı .R/ j f .0/D 0g also has full image in QI.R/.Let PL ı;C .R/ D ff 2 PL ı .R/ j f .x/D x; x Ä 0g; PL ı; .R/ D ff 2 PL ı .R/ j f .x/D x; x 0g: The group of quasi-isometries of the real line cannot act effectively on the line absolute values.We will implicitly use this fact in the following context.As PL ı .R/ has a full image in QI.R/ (by Sankaran [9]), we take representatives of quasi-isometries which are homeomorphisms in the rest of the article.Proof For any f 2 W .R/ and sufficiently large x > 0, its derivative satisfies that jhf h 1 .x/ 0j D j.e f .lnx/ / 0 j D j.xe f .lnx/ ln x / 0 j D je f .lnx/ ln x .1 C f 0 .lnx/ 1/j D je f .lnx/ ln x f 0 .lnx/j Ä e sup x2R jf .x/xj sup x2R jf 0 .x/j: The case for negative x < 0 can be calculated similarly.This proves that hf h 1 is a quasi-isometry.
The following result was proved by Sankaran [9].
Proof For any f 2 Diff Z .R/, we have f .xC 1/ D f .x/C 1 for any x 2 R.This means sup x2R jf .x/xj < C1.Since f .x/x is periodic, we know that f 0 .x/ is bounded.Suppose that f .x/> x for some x 2 OE0; 1.Take y n D e xCn for n > 0. Let h be the function defined in Lemma 2.3.We have jhf h 1 .yn / y n j D je f .xCn/e xCn j D je f .x/e x je n !1; Proof Since H is normal, it's enough to prove that Diff Z .R/ \ H D feg, the trivial subgroup.Actually, for any f 2 Diff Z .R/, the conjugate hf h 1 is a quasi-isometry as in the proof of Corollary 2.4.If hf h 1 2 H , then x!1 e f .lnx/ ln x D 1: Since f .x/x is periodic, we know that f .lnx/ D ln x for any sufficiently large x.
But this means that f .y/D y for any y, so f is the identity.
2.2 Affine subgroups of QI.R/ Lemma 2.6 The quasi-isometry group QI.R C / (actually, the semidirect product for x 0. We define A t .x/D B i;s .x/D x for x Ä 0. Since the derivatives are bounded for sufficiently large x, we know that A t and B i;s are quasi-isometries.For any x 1, .x/: For any x 1, and ˇÄ js 1 s 2 j by Newton's binomial theorem.This means that B i;s 1 B i;s 2 and B i;s 1 Cs 2 are of bounded distance.It is obvious that When i < j are distinct natural numbers, for any x 1.This proves that images OEA t ; OEB i;s 2 QI.R 0 / satisfy the relations.By abuse of notation, we still denote the classes by the same letters.
We prove that the subgroup generated by fB i;s j i 2 R 1 ; s 2 Rg is the infinite direct sum L i 2R 1 R. It's enough to prove that B i 1 ;s 1 ; B i 2 ;s 2 ; : : : ; B i k ;s k are Z-linearly independent for distinct i 1 ; i 2 ; : : : ; i k and nonzero s 1 ; s 2 ; : : : ; s k 2 R.This can directly checked.For integers n 1 ; n 2 ; : : :

Left-orderability
The following is well known; for a proof, see [7, Proposition 1.4]: Lemma 3.1 A group G is left-orderable if and only if , for every finite collection of nontrivial elements g 1 ; : : : ; g k , there exist choices " i 2 f1; 1g such that the identity is not an element of the semigroup generated by fg " i i j i D 1; 2; : : : ; kg.
We assign " i D 1 for the first case and " i D 1 for the second case.For the third case, let is not empty.We choose another sequence fx 2;k g such that sup k2N jf i 0 .x2;k / x 2;k j D 1. Similarly, after passing to a subsequence, we have for each f 2 S 1 that either f .x2;k / x 2;k !C1, f .x2;k / x 2;k ! 1 or sup k2N jf .x2;k / x 2;k j Ä M 0 for another real number M 0 .Assign " i D 1 for the first case and " i D 1 for the second case.Continue this process to define S 2 ; S 3 ; : : : and choose sequences fx i;k g; i D 3; 4; : : : to assign " i for each f i .Note that the process will stop at n times, as the number of elements without assignment is strictly decreasing.

Lemma 3.2
The group QI.R C / is not locally indicable.
Proof Note that QI.R C / contains the lift z of PSL.2; R/ < Diff.S 1 / to Homeo.R/ (Corollary 2.4).But this lift z contains a subgroup D hf; g; h W f 2 D g 3 D h 7 D fghi, the lift of the .2;3; 7/-triangle group.There are no nontrivial maps from to .R; C/; for more details see [2, page 94]. 4 The quasi-isometric group cannot act effectively on the line The affine group R >0 Ë R cannot act effectively on the real line R by homeomorphisms with A t a translation for each t.
Proof Suppose that R >0 Ë R acts effectively on the real line R with each A t a translation.After passing to an index-2 subgroup, we assume that the group is orientation-preserving.If B 1 acts freely on R, then it is conjugate to the translation x/ C 1 for any x.Since A 1 2 maps intervals of length 2 to an interval of length 1, it is a contracting map, and thus has a fixed point.
If B 1 has a nonempty fixed point set Fix.B 1 /, choose I to be a connected component of R n Fix.B 1 /.Suppose that A 2 .x/D x C a, a translation by some real number a > 0. Since A 2 D A n 2 1=n , we have A 2 1=n .x/D x C a=n for each positive integer n.For each n, let x a=n/ C a=n for any x 2 R, we know that F n .I / D I for sufficiently large n.Without loss of generality, we assume that I is of the form .x; y/ or .1; y/.Choose a sufficiently large n such that y a=n 2 I .We have which is a contradiction to the fact that F n .I / D I .The following is similar to a result proved by Militon [6].
Lemma 4.3 (Militon [6]) Let D PSL 2 .R/ and z < Homeo C .R/ be the lift of to the real line.Any effective action W z ,! Homeo C .R/ of z on the real line R is a topological diagonal embedding.
Proof After passing to an index-2 subgroup, we assume the action is orientationpreserving.Let W R !R be the translation x 7 !x C 1. Suppose that Fix. .// ¤ ∅.Note that lies in the center of z .The quotient group D z =h i acts on the fixed point set Fix. .//.For any f 2 and x 2 Fix. .//, we denote the action by f .x/without confusion.Choose any torsion-element f 2 and any x 2 Fix. .//.We must have x D f .x/,for otherwise x < f .x/< f Proof If is injective, the previous lemma says that is a topological diagonal embedding.Therefore, .A/ is continuous.
We will need the following elementary fact.the restriction map j R is R-linear for each direct summand R.This is a contradiction to the fact that is injective.Therefore, the group G cannot act effectively.
Proof of Theorem 1.

Lemma 4 . 1
The following was proved by Mann[4, Proposition 6].Consider the affine group R >0 Ë R, generated by A t and B s for t 2 R >0 and s 2 R satisfying

Definition 4 . 2 A
topologically diagonal embedding of a group G < Homeo.R/ is a homomorphism W G ! Homeo C .R/ defined as follows.Choose a collection of disjoint open intervals I n R and homeomorphisms f n W R !I n .Define by .g/.x/ D f n gf 1 n .x/when x 2 I n and .g/.x/ D x when x … I n .

Lemma 4 . 5
Let W .R; C/ !.R; C/ be a group homomorphism.If is continuous at any x ¤ 0, then is R-linear.Proof For any nonzero integer n, we have .n/D n .1/and .1/D 1 n n D n 1 n .Since is additive, we have m n D m 1 n D m n .1/for any integers m; n ¤ 0.
2.1 QI.R C / is not simple Let QI.R C / be the quasi-isometry group of the half-line OE0; C1/.Note that the quasiisometry group QI C .R/ D QI.R C / QI.R / and QI.R C / Š QI.R /, by Lemma 2.1.Let H D fOEf 2 QI.R C / j lim x!1 .f.x/x/=x D 0g.Theorem 1.1 follows from the following theorem.Theorem 2.2 H is a proper normal subgroup of QI.R C /.In particular, QI.R C / is not simple.Therefore, OEg 1 fg 2 H . It's obvious that the function f defined by f .x/D 2x is not an element in H .The function defined by g.x/ D x C ln.x C 1/ gives a nontrivial element in H . Thus H is a proper normal subgroup of QI.R C /.
H by the following construction.Let a t ; b i;s W R !R be defined by a t .x/D x C ln t and b i;s .x/D ln.e x C se >0 , i 2 R 1 and s 2 R. It can be directly checked that a t 2 Diff Z .R/ and b i;s 2 W .R/ (defined in Lemma 2.3).Let h.x/ D e x .A direct calculation shows that ha t h 1 D A t and hb i;s h 1 D B i;s , as elements in QI.R C /.
[4, proof of Theorem 1.2 follows a similar strategy used by Navas to prove the left-orderability of the group G 1 of germs at 1 of homeomorphisms of R; cf [2, Remark 1.1.13]or[4,Proposition2.2].Proof of Theorem 1.2 It's enough to prove that QI.R C / is left-orderable.Let f 1 ; f 2 ; : : : ; f n 2 QI.R C / be any finitely many nontrivial elements.Note that any 1 ¤ OEf 2 QI.R C / has sup x>0 jf .x/xj D 1.This property doesn't depend on the choice of f 2 OEf .Without confusion, we still denote OEf by f .Choose a sequence fx 1;k g R C such that sup k2N jf 1 .x1;k / x 1;k j D 1.For each i > 1, we have either sup k2N jf i .x1;k / x 1;k j D 1 or sup k2N jf i .x1;k / x 1;k j Ä M for a real number M .After passing to subsequences, we assume for each i D 1; 2; : : : ; n that either 2.x/ < < f k .x/foranyk.Since is simple, we know that the action of z on Fix./ is trivial.For each connected componentI i R n Fix..//,weknow that j I i is conjugate to a translation.The group D z =h i acts on I i =h ./i D S 1 .A result of Matsumoto [5, Theorem 5.2] says that the group is conjugate to the natural inclusion PSL 2 .R/ ,! Homeo C .S 1 / by a homeomorphism g 2 Homeo C .S 1 /.Therefore, the group .z /j I i is conjugate to the image of the natural inclusion z ,! Homeo C .R/.For any injective group homomorphism W z !Homeo.R/, the image .A/ is a continuous one-parameter subgroup; ie lim a!a 0 .ta / D .t a 0 / for any a 0 2 R.
For a real number a 2 R, lett a W R !R; x 7 !xC abe the translation.Denote by A D ht a W a 2 Ri, the subgroup of translations in the lift z of PSL 2 .R/.Corollary 4.4 3 Suppose that QI C .R/ acts on the real line by an injective group homomorphism W QI C .R/ !Homeo.R/.The group QI C .R/ contains the semidirect product R >0 Ë L i 2R 1 R , by Lemma 2.6.The subgroup R >0 (as the image of the exponential map) is a homomorphic image of the subgroup R < z , which is the lift of SO.2/=f˙I 2 g < PSL 2 .R/ to Homeo.R/.Note that z is embedded into QI C .R/ (see Corollary 2.4 and its proof).By Lemma 4.3, any effective action of z on the real line R is a topological diagonal embedding.This means that the action of R >0 is a topological diagonal embedding (Corollary 4.4).Theorem 4.8 shows that the action of R >0 Ë L i 2R 1 R is not effective.