Coarsely geodesic metrics on reductive groups (after H. Abels and G. A. Margulis) Текст научной статьи по специальности «Математика»

Математические структуры и моделирование 2005, вып. 15, с. 5-17

УДК 512.817

COARSELY GEODESIC METRICS ON REDUCTIVE GROUPS (AFTER H. ABELS AND G. A. MARGULIS)

G.A. Noskov

В статье изучаются свойства функций длины на группах.

We are going to study the length functions on a group G, that is the functions ^ —> |p| : (f? —> K., satisfying the following axioms:

• Positivity: 111 = 0 and \g\ > 0 for all nonidentical g Є G;

• Triangle inequality: \gh\ < \g\ + \h\, g,h Є G;

• Symmetry: |#| = |g~l\} g Є G.

d& J~

Any length function gives rise to a left invariant metric d on G as usual: d(g, h) =

d= \g~1h\. And conversely, any left invariant metric defines a length function.

There are plenty of left invariant Riemannian metrics on any connected real Lie group G, although all of them give rise to a Lipshitz equivalent distance functions. On the other hand if 0 is a compact symmetric neighbourhood of identity 1 G G then we can define a word length function on G by |g| = тіп{г Є Z+ | g Є 0*}. One can easily see that d is quasi-isometric to the metric induced by any left invariant Riemannian metric on G. Thus the problem of classification of metrics of above type up to a quasi-isometry is trivial - any two of them are quasi-isometric. Both two classes of metrics, introduced above, have a common feature - they are «coarsely geodesic» (see below the definition).

Recently H. Abels and G. A. Margulis gave much more refined classification of coarsely geodesic left invariant proper metrics on reductive Lie groups up to coarse equivalence [1]. They have defined a class of so called normlike pseudometrics on a reductive group, and have proved the following theorem.

Theorem 1. Let G be a reductive Ж-group and G = G(M)°. Then any left invariant coarsely geodesic proper metric is bounded distance away from a unique normlike pseudometric on G.

Copyright © 2005 G.A. Noskov.

Omsk Branch of Institute of Mathematics, and Mathematisches Institute der Heinrich-Heine-Universitat Duesseldorf.

E-mail: noskov@private.omsk.su

6

G.A. Noskov Coarsely geodesic metrics on reductive groups...

This theorem gives the picture of a metric «in a large scale» and favorably compare to the following result of V . Berestovsky which is «local» in nature [2] (for example it does not apply to word metrics):

Theorem 2. Any interior left invariant metric on a Lie group is the Carnot-Caratheodory-Finsler one.

We give an exposition of Abels & Margulis result, sacrificing some generality, but not the ideas.

1. Coarse world

Definition 1. A parameterized curve (not necessarily continuous!)

a{t) : [0, t0] —> A

c

in a metric space (X, d) is called a C-coarse geodesic, C > 0 if d(a(s), a(t)) = \s — t\

c

for all s, t Є [0, to]- (The symbol = means equality up to an error not exceeding C.) The space (X,d) is called (7-coarsely geodesic, if any two points x,y Є X can be connected by a ( '-coarse geodesic.

A map / : X —> Y of metric spaces is (A, £>)-uniform, A, В > 0 if

V x, x' Є X : d(x, x') < A => d(f(x), fix')) < В

and / is called uniform if V A > 0 there is В > 0, such that / is (A, £>)-uniform. For example, any ( '-coarse geodesic is (В. В + ( ')-uniform for any В > 0.

A map / : X —> Y of metric spaces is proper if the preimage of a bounded set is bounded.

We say that the metrics di,d2 on a space X are Hausdorff or coarsely equivalent iff \d\ — d2\ is a bounded function on X x X.

2. Reflection groups

We recall notation and facts about finite groups generated by reflections and which we use later on. Sufficient references for this are [3,4].

Let В be a Euclidean space, i.e. a finite dimensional vector space with an inner product. The reflection in a hyperplane H is the linear transformation s# : V —> V which is the identity on H and is multiplication by -1 on the (one-dimensional) orthogonal complement IIі. A finite reflection group is a finite group W of linear transformations generated by reflections. We call the fixed-point subspace Vo = Vw the inessential part of V, and its orthogonal complement V\ the essential part of V. Let 7Ї denote the set of all reflecting hyperplanes H with in s# Є W. For a reflection s e S, we denote its reflecting hyperplane by IIs. The connected components of the

complement of the union (J H in Rn are called Weyl chambers, and the closure of

неп

a Weyl chamber is called a closed Weyl chamber. Any closed Weyl chamber A+ is a fundamental set for W in K". i.e. each fF-orbit has precisely one point in A+.

Математические структуры и моделирование. 2005. Вып. 15.

7

In particular we have a Coxeter projection c : Rn —» A+. It implies that W acts simply transitively on the set of Weyl chambers or, equivalently, any closed Weyl chamber can be uniquely represented as wA+, w Є W. The hyperplane Ls defines two closed halfspaces of M" and those one, containing A+, we denote by Lfi. We fix a IT-invariant inner product ( ,) on M" . For each reflection s there is a unique unit vector as, pointing to Lf and orthogonal to Ls and we call this vector an s-root. In terms of roots A+ = {x Є Rn | (x, as) >0 V s Є So}.

Any Weyl chamber is a unique irreducible intersection of halfspaces and we call the corresponding hyperplanes the walls of the chamber. The set So of all reflections in the walls of A+ generates IT. The vectors {as \ s Є So} form a basis of V\. Therefore Vfi A A+ is a simplicial cone of dimension | So |, and we have an orthogonal decomposition A+ = Vo + (Tl П A+). We also recall that the angle between any two roots, corresponding to different walls of A+ is obtuse.

We call convex cone A++ spanned by {as \ s Є So} the dual of A+. The linear functional £ on V is called positive with respect to A+ if it takes nonnegative values on A++. Let Wx = {w Є W \ wx = x} denote the stabilizer of a point x Є Rn in IT. It is known that Wx is generated by Wx П So for any x Є A+.

Lemma 1. (Supporting functionals) Let || • || be a W-invariant norm on V. For any nonzero x Є V one can find a linear functional £x on Rn such that

\\£\\ = 1, £x(x) = \\x\\,

and ix is positive with respect to any Weyl chamber containing x.

Proof. It follows from the Hahn-Banach theorem that there exists a linear functional £x on Rn such that ||4|| = 1 and £x(x) = ||x||. Averaging lx over the stabilizer Wx of x in IT and using IT-invariance of || • ||, we may assume £x to be invariant under Wx. Fix a Weyl chamber A+, containing x. It remains to prove that £x is positive with respect to A+, that is £x(as) > 0 for every s Є So, then by invariance £x(as) = 0. If sx Ф x, then

£x{sx) < ||4|||M| = ||x|| =£x(x),

i. e. £x(x) — £x(sx) is positive. Moreover, x — sx is a positive multiple of as and therefore £x(x) — £x(sx) is a positive multiple of £x(as), hence £x(as) is positive and thus £x is positive with respect to A+. Since £x is ITr-invariant we conclude that £x is positive with respect to any wA+,w Є Wx. It is easy to see that any Weyl chamber, containing x is of the form wA+ for some w Є Wx. U

3. Length functions on Rn and stable norms

Let I • I be a length function on additive group Rn. If а Є Rn then by the triangle inequality the sequence \ma\,m Є N is subbadditive and therefore the following limit

і и def a = hm

ma

m^oo rn

G.A. Noskov Coarsely geodesic metrics on reductive groups...

exists for every а Є Rn and clearly |a| > ||a||. We call ||a|| a stable norm of a. It follows from definition that the stable norm is nonnegative, symmetric, homogeneous, satisfies triangle inequality, but it might happen that the stable norm is not positive.

Definition 2. A length function | • | on Rn is proper, iff both | • | and any Euclidean metric on Rn have the same system of bounded subsets.

The definition is stronger that that of given in [1], but this does not affect the results we are going to prove.

We need a criterion for a stable norm to be positive. For example it is so if the length function is coarsely geodesic. We wish to strengthen this remark as follows:

Lemma 2. (When the stable norm is a norm) Suppose Rn is given with a proper length function \ • \ which is «С-uniform» for some C > 0 in a sense that for each а Є Rn there is a (1, C) -uniform curve a : [0, |o|] —> Rn, starting at 0 and ending distance at most C from a. Then the associated stable norm || • || is a norm.

Proof. It follows easily from properness of the metric that condition of lemma holds for standard Euclidean length function | • |e with some constant, say C. In particular for each а Є Rn,m Є N there is a (1, C")-uniform (with respect to | • |e) curve a : [0, |a|] —» Rn, starting at 0 and ending distance at most C from та. The sequence of inequalities

, [l"41

m|a|e = |ma|e = |a(|ma|)|e < |«(г) — а (і — l)|e + C' < C'/([|ma|] + 1)

І—1

implies that if a is nonzero, then \ma\ grows at least linearly with m for nonzero a, hence || a 11 is positive. ■

4. Groups with Cartan projection and stable norms

Definition 3. Suppose that we are given:

1) A group G with a coarsely geodesic length function | • |;

2) Its subgroup A with a fixed isomorphism A ~ Rn;

3) Action of a reflection group W on A and a Weyl chamber A+.

We say that the map a(g) : G —> A+ is a Cartan projection if the following conditions are satisfied:

CPI) The restriction of the map a(g) to A is a Coxeter projection;

CP2) The map a(g) coarsely preserves the norm | • |;

CP3) «Triangle inequality»: a(g) + a(h) — a(gh) Є A++ for any g,h Є G.

We will call an assembly Q = (G, \ ■ |, A, IV, A+, a(g)) a group with Cartan projection. We fix such one throughout this section. Equally with the above notation, we write sometimes, a(g) = ag for the Cartan projection. Clearly, for any w Є IT the map wa(g) : G —> wA+ is again a Cartan projection.

Lemma 3. If the length function \ • \ is proper on A then the Cartan projection is uniform.

Математические структуры и моделирование. 2005. Вып. 15.

9

Proof. Note that

dgh ^ ^h A

and

Cigh dg Є &h~1 T A

(by Axiom CP3). By properness ah and ah-1 are in a bounded subset of 1C, hence so is the intersection of the right hand sides, since

(a-A++)n(b + A++)

is compact for any а, Ь Є 1C and hence so is the intersection of the right hand sides. It follows that \agh — agI is bounded whenever \h\ is bounded, that is the Cartan projection is uniform. ■

By construction of Section 3. associated to the length function on G is the stable norm on Л ~ I" and we are interested when it is positive.

Lemma 4. The stable norm is positive on A.

Proof. Fix nonzero a Є A and let g(t), 0 < t < |ma| be a coarse geodesic in G from 0 to ma. Note that any coarse geodesic is (1, Cj-uniform for some C > 0. Since Cartan projection is uniform, it follows that the projected curve ag(t) is uniform again. It follows that ||a|| is positive by Lemma 2. ■

Lemma 5. The stable norm on A is W-invariant.

Proof. Since the restriction of Cartan projection to A is a Coxeter projection, and it coarsely preserves the norm, we see that the restriction of length function to A is coarsely W-invariant. Hence the stable norm is IF-invariant. ■

Definition 4. We extend a stable norm onto the whole G via Cartan projection: ||р 11 = ag,g e G. Similarly, if £ is a functional on IRC, we extend it onto G by precomposing with Cartan projection.

Lemma 6. Stable norm on G satisfies the triangle inequality:

MI<IMI + INI

for any g,h Є G. Moreover for any functional £ on 1C, positive with respect to A+, we have

£(gh) < £(g)+£(h)

for any g,h Є G.

Proof. Triangle inequality for £ immediately follows by application of £ to the inclusion in the axiom CP3 above. Let g,h Є G. If \\agh\\ is zero then the assertion is obvious, so we assume that it is nonzero. According to Lemma 1, there exists a linear functional £ on 1C such that ||£|| = 1, £{agh) = Ца^Ц, and £ is positive with respect to A+. Then applying triangle inequality for £ we obtain

10

G.A. Noskov Coarsely geodesic metrics on reductive groups...

5. Miraculous elements

Let Q = (G, I • I, A, W, A+, а(д)) be a group with Cartan projection and let Co be a constant, such that the length function is Go-coarsely geodesic and the Cartan projection Co—coarsely preserves the length function.

Definition 5. We say that а Є Шп is C-good, C > 0, if ||a|| > |a| — C. We say that a linear functional t is almost supporting at a if ||Z|| = 1 and £fia) > ||a|| — 1.

Our aim is to produce C'-good elements д in «every direction».

Lemma 7. There is a constant C depending only on Q and on natural щ such that for any а Є A+ with ||a|| > 1 and for any finite family C of linear functionals on Rn coarsely supporting at a, positive with respect to A+ and of cardinality at most no there is an element g Є G such that,

і с и I

aJ = a

and £(ag) > \\a\\ — C, £ Є C.

(1)

Proof. Let g(t) : [0, to] —► G be a Go-coarse geodesic in G from 0 to та, in particular to = |ma|. We define

, t0 . tj = —J

m

for j = 0,..., m and we define

9j = g(tj-1) 1g(tj),

for 1 < j < m. We have та = gi • • • gm and

I I def at \ 14 w Co to Cq/п

\9j\ — d(g(tj-i),g(tj)) — — —

for all j. In particular

Cq і і Cq j J 1

I I I

a9j I — \9j\ —

= a

m

for m 0 and we conclude that

та

m

dg

2Cq+1 j j ^2 j j

for all j and rn У> 0 (take into account that the composition of relations =,= is

Cfi-D

= ). Thus, any Qj for rn 0 fulfils the first condition in the Lemma. Further, for

£eC

£(§j) ^ lla9j ll ^ \ag, \ — \dj \ + Go < ||a|| + Go + 1 (2)

for m T> 0. Since any £ is coarsely supporting and by triangle inequality from Lemma 6 we have

Математические структуры и моделирование. 2005. Вып. 15.

11

We wish to derive from 2 inequalities above that

t(9j) > IMI - C

for all £ Є jC and universal constant C. For any C and any £ Є C we easily conclude from (2),(3) that

t, def JJ_ ( • I 1 ^ ^ Щ \ ^ || || f~i~\ ^ Tfl(Cо T 2)

be,m = #{j | 1 < J < m, e{gj) < ||o|| - G) < , n , 1 •

Oo -г О + I

Thus, taking С > (щ + l)(Co + 2) we obtain that

bi,m 1 m щ + 1

for any £ Є jC and for sufficiently large m. From this and the definition of b( rn we deduce the existence of j, 1 < j < m, such that for all £ Є C, £(gj) > ||a|| — C and we take g = gj. U

6. Cartan projections of geodesic curves

Let Q = (G, I • I, A, W, A+, a(g)) be a group with Cartan projection and let Co be a constant, such that the length function is Со-coarsely geodesic and the Cartan projection Co—coarsely preserves the length function. Our aim is to construct a continuous curve, consisting entirely of C-good elements and along which the given finite family of functionals grow coarsely with time.

Lemma 8. There exists a constant C > 0, depending only on Q and on a natural number щ such that for any а Є A+ and an arbitrary family C of coarsely supporting

at a functionals, positive with respect to A+, and of cardinality at most щ there exists

c

a continuous curve a : [0, to] —>■ A+,to = |a| starting at 0, such that

\a(t)\ = t and £(a(t)) > t — C, t Є [0,to\, £ Є C. (4)

Proof. Let D be a constant exceeding both Co and the constant given by Lemma 7. In particular, there is an element g Є G such that

|ofl| = ||a|| and £{ag) > ||a|| — D, £ Є C. (5)

Let g(t) : [0, to] —>■ G, be a D-coarse geodesic in G connecting 1 and g, in particular t0 = |g|. Since the Cartan projection coarsely preserves the length and gif) is a

H-coarsely geodesic we conclude that |as(t)| = |g(t)| = t and thus \ag(t)\ == t, hence the curve os(t) satisfies the first assertion of the Lemma with C = 2D.

Let g'it) = g(t)~1g, so that g = g{t)g'(t). For £ Є C we have

INI - D < £(g) < £(g(t)) +£(g'(t)) < £(g(t)) + \ag/{t)\ = £(g(t)) + \g'(t)\ =

12

G.A. Noskov Coarsely geodesic metrics on reductive groups...

= + d(g(t),g(t0)) = £{g(t)) + t0-t.

The equalities \g\ = ||a|| and to = \g\ imply to 2= ||a||, and substituting this to the above we get

£(g(t)) >t — 5D, 0 < t < to,

thus the curve ag(t) satishes the second assertion with C = 5D.

To make ag(p continuous we define the curve a(t) as the curve which coincides with ag(t) at the integral moments j, 0 < j < [to] and at the moment to and which is linear on the segments \j,j + 1], 0 < j < [to] — 1, [[t0], to] • First note git) and

a(g) are uniform maps, so there is a constant E depending only on Q, such that that |o:(f) — <u([t])| < E,t Є [0, to]- Let C = E + 5D + 1. We have

||a:(f)| — t| < |a:(t) — a([f])| + ||a([t])| — t\ < E + 2D,

c

hence \a(t)\ = t for all t Є [0,to]- (We ask the reader to forgive the conflict of notations for length function on G and for absolute value for reals). Finally, for any t Є [0, to] and any £ є C we have

£(a(t)) = ^(«([t])) + £(a(t) — a([t])) > [t] — 5D — |a(t) — a([t])| > t — E — 5D — 1.

7. Main Theorem

Theorem 3. Let Q be a group with Cartan projection. Suppose that the length function is proper on G. Then

sup|M - 1ЫН < oo.

g&G

In particular any left invariant, coarsely geodesic, proper metric on G is bounded distance away from a unique normlike pseudometric on G.

Proof. The main case is g Є A ~ Rn, from which the general case follows easily by definition of the stable norm on G. Since our length function is proper, || • || is a norm. Fix a real r > 0 and let Br C A be a unit ball of radius r about origin. In this proof we define a continuous map <p : Br —» Rn, such that <p(Br) consists entirely of (7-good points and the image contains B0j-c. The last claim is proved using a topological argument, namely the degree of maps between spheres.

Applying standard arguments, one can show that there exists a triangulation T of the boundary dBr such that each simplex of T has || • ||-diameter <1/2 and lies entirely in some Weyl chamber. Let % denote the barycentric subdivision of T. For any simplex cr in either T or %, we denote by Va the set of vertices in cr. Let ox denote the smallest simplex in T which contains a given x Є dBr.

According to Lemma 1, for any nonzero igK", one can find a linear functional £x on Mn such that \\£x\\ = 1, £x(%) = IMI> and £x is positive with respect to any

Математические структуры и моделирование. 2005. Вып. 15.

13

Weyl chamber containing х. Note that £x is almost supporting at any point of ax. Indeed, for у Є <jx we have

£x(y) = INI +4(у ~x)> \\y\\- 2||у - x\I > ІІ2/Ц - 1.

For each vertex v Є % we define the set of functionals

£-v = {4 : u ^ VUv}.

Note that this set satisfies the conditions of the Lemma 8. Indeed, the functionals are of norm 1, they are almost supporting at v by previous remark, and if wA+ is any Weyl chamber, containing v, then the simplex av is contained in wA+ too, hence all functionals are positive with respect to wA+. Now by Lemma 8 for some C > 0, depending only on Q we find a continuous curve a : [0, tv\ —> Rn, such that

|cq,(£)| = t, 0 < t < 4, tv = ||u||, a(0) = 0,

and

£u{oLv(t)) >t — C for any u Є VUv.

We now define a continuous map у : II, Rn as follows. Take any z Є dBr, and let a be a simplex in % which contains z. Represent z as a convex linear combination

Z ^ 'j AV,Z^1

vGV (a)

and define

<p(sz) = ^2 w(sAVjZtv), 0 < s < 1. vev„

It is clear that the nonzero coefficients Xv>z in the decomposition above do not depend on the choice of a. Therefore y(tz) does not depend on the choice of a either. Since the curves {oy(s)} are continuous, the map у : Br —» Rn is continuous.

Claim. y(Br) consists of C-good points.

Since % is the barycentric subdivision of T, one can find a vertex u in a such that u Є VGv for all v Є Va. Then it follows

IN44II > 4(<p(sz)) = ^2 tu{oLv{s\v,zav)) > J2 sXv,zav - C>

v^V(a) vGVo-

^ ^ II С ^ I ^ ^ olv(sA^^qn)I C —

VeVa VeVa

Let Na},A Є [0,1] be a linear homotopy between identity and ф. It follows from inequalities above that

INaNII > 4Na40) = 4(Ay(z) + (1 - A)z) > r — C for any z Є dBr and 0 < A < 1. We get from above that

Br~c~і П Lp\(dBr)

for any 0 < A < 1. The following topological lemma finish the proof.

14

G.A. Noskov Coarsely geodesic metrics on reductive groups...

Lemma 9. Suppose that a continuous map f : Br —> Rn satisfies the following property: t Br_c~і П f\(dBr)) for any 0 < Л < 1, where Д is a linear homotopy between identity and f. Then the image f(Br) contains Br_c~і for ad r sufficiently large.

Proof. Arguing by contradiction, suppose that f(Br) does not contain a point у Є Br_c~і- By assumption the restrictions of Id and / to dBr are homotopic as maps into Rn — {y}. Composing with projection map of the last space onto the sphere Sr, we obtain that the identity map of the sphere is homotopic to the constant map - this contradicts to the well known fact (the continuous maps of the sphere into itself of different degree are not homotopic). This contradiction proves the Lemma. ■

Theorem is proved. ■

8. Examples of groups with Cartan projection

The notion of a group with a Cartan projection is motivated by the classical Cartan decomposition for semisimple or, more generally, reductive Lie groups. Let

G = G(R)°

be the connected component of the identity of the group G(R) of R-rational points of a reductive R-group G. Fix a proper coarsely geodesic length function | • | on G. Let A be a maximal R-split torus in G and A = A(R)°. The group A is isomorphic to Rn where n = dim A. Let A+ C A be a Weyl chamber for the Weyl group W =Mg{A)/Zg{A).

Theorem 4. The assembly Q = (G, \ • |, A, IV, A+, a(g)) is a group with Cartan projection.

Proof. It is well known that G admits a Cartan decomposition G = KA+K, where К is a suitable maximal compact subgroup of G and that associated «Cartan projection»

a(g) : G —» A+

is well defined. Moreover, if -w Є IT and а Є A+ then w{a) Є KaK, and this implies that the restriction of a(g) onto A is a Coxeter projection, thus CPI) is fulfilled. Since К is compact and the length function is proper, it follows that the Cartan projection coarsely preserves the length function, hence CP2).

The axiom CP3) is rather nontrivial and relies on the presentation of positive linear functionals as linear combinations of highest weights of rational representations [5]. Let 7Г : G —► GL(V) be a rational representation of the group G defined and irreducible over R. We decompose V into the direct sum

V= фи. U^{0}. (6)

ХЄХ(А)

Математические структуры и моделирование. 2005. Вып. 15.

15

of the weight spaces

Vx = {v Є V І 7Г{a)v = x(a)v for any a ^ -4},

where X(A) is the group of rational characters of A. Let Є X(A) be the highest weight of the representation %. Then

X(a) < pn(a) if а Є A+ and Vx ф {0}. (7)

It is well known that one can introduce a 7t(K)-invariant inner product on V with respect to which the transformations тг(а),а Є A, are self-adjoint. Then the subspaces Vx in the decomposition (6) are mutually orthogonal, and it follows from

(7) that Цтг(^)|| = pn(a(g))j f°r any 9 Є G, where the norm is taken with respect to the inner product just defined. As a consequence, we get that for any g,h Є G,

logpw(a(gh)) < logрфа(д)) + log pw(a(h)). (8)

It is well known that any linear functional £ on A = Rn, which is positive with respect to A+, can be represented as a positive linear combination

m

t = J^Mog/b^, bi > 0,

i—1

where 7Tj, 1 < і < m, are rational representations of C, defined and irreducible over M. Then it follows from (8) that for any g,h Є G,

£(a(gh)) < i(a(g)) + £(a(h)). (9)

Since і is an arbitrary positive functional we get from this and the definition of A++ that the axiom CP3) is satisfied. ■

9. Generalizations, questions, problems

The notion of a group with a Cartan projection used above is not enough to treat the case of reductive groups over local fields. The generalization is given in [1] and here we give a sketch. Namely we must allow not only Rn but any closed cocompact group D of Rn. An action of a reflection group W on Rn should leave D invariant and there must be a compact symmetric set M C Rn, containing 0, such that for each w Є W there is an inclusion

wA+ C [D П wA+) + M.

We say that the map

a(g) : G —> A+

is a Cartan projection if CPI), CP2) are satisfied and CP3) is satisfied in the following relaxed form:

п(р) T o(/i) — a(gh) G T M

16

G.A. Noskov Coarsely geodesic metrics on reductive groups...

for any g,h Є G. The generalization includes the case of invariant metrics on Z" as well as invariant metrics on Rn, but seems it does not include the case of Zn-invariant metrics on Rn, which were treated by D. Burago [6].

Question 1. Is the analog of Abels-Margulis theorem for lattices in a reductive group valid? (Presumably not). The same question for the metrics on a reductive group invariant under translations by the elements of the lattice. (The answer is presumably yes).

Question 2. What is the relation of the triangle inequality for stable norm to the Gelfand-Naimark theorem about singular values of the product of matrices?

Question 3. What is the analog of Abels-Margulis theorem for nilpotent, solvable, general Lie groups? Look at the S. Krat paper.

Question 4 What is the structure of the asymptotic cone of a reductive group with a normlike pseudometric? Presumably they are Minkowski buildings. The question is related to the results of Kleiner-Leeb, Thornton, Parreaut.

Question 5 What is the relation between Abels-Margulis and Berestovsky theorems? For example, could one calculate a normlike pseudometric associated to a Carnot-Caratheodory-Finslerian metric on a reductive group?

Question 6. Let M be a set provided with two interior metrics d\ and Т>. Assume that a group G acts cocompactly on M by isometries with respect to both metrics and

V di (x,y) ,

lim —--------- = 1.

d2(x,y)^oо 0,2 [X, у)

Due to a result of D. Burago [6], if G = Zn, then d\, d2 are coarsely equivalent. This fact means that all metrics on M diverge linearly or stay within a finite distance from each other. Burago raised the question for which groups the same statement could be true. The Abels-Margulis result easily implies the positive answer for metrics on reductive groups. D. Burago suggested two different directions. The first is the case of semi-hyperbolic groups, i.e., groups of isometries of a space whose curvature is bounded from above by 0. The other one is the case of nilpotent groups and first of all, the Heisenberg group. In some cases the fact that two metrics cannot diverge more slowly than linearly could be described as the finiteness of the Gromov-Hausdorff distance between the group with induced metric and its asymptotic cone. In the case of the abelian group Zn the asymptotic cone is Rn and it lies within a finite Gromov-Hausdorff distance from Zn. The Gromov-Hausdorff distance between Heisenberg groups and its asymptotic cone is finite [7].

References

1. Abels H., Margulis G.A. Coarsely geodesic metrics on reductive groups // Preprint, Bielefeld, 2003.

2. Berestovskii V.N. Homogeneous manifolds with an intrinsic metric.II // Siber. math. J. 1989. V. 30. P.180-191.

3. Brown K.S. Buildings. Springer Monographs in Mathematics. Berlin, Heidelberg: Springer-Verlag, 1996.

17

4. Humphreys J.E. Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics, 29. Cambridge: Cambridge University Press, 1990.

5. Vinberg E.B. Lie groups and Lie algebras, III. Springer-Verlag, 1994.

6. Burago D. Perodic Metrics // Advances in Soviet Mathematics. 1992. V. 9. P.205-210.

7. Krat S.A. Asymptotic properties of the Heisenberg group // Zapiski Nauchnykh Sem-inarov S.-Peterburg. Otdel. Mat. Inst. Steklov. (РОМІ). Geom. і Topol. 1999. V.261, №. 4. P.125-154.