QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES AND LATTICES: REDUCIBLE CASE

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1 QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES AND LATTICES: REDUCIBLE CASE THANG NGUYEN Abstract. We study quas-sometrc embeddngs of symmetrc spaces and non-unform rreducble lattces n semsmple hgher rank Le groups. We show that any quas-sometrc embeddng between symmetrc spaces of the same rank s decomposed as a product of quas-sometrc embeddngs nto rreducble symmetrc spaces. We thus can extends the rgdty results n [FW18] and [FN] to the settng of semsmple Le groups. We also present some examples when the rgdty does not hold, ncludng frst examples n whch every flat s mapped nto multple flats. 1. Introducton The coarse geometry of spaces have been capturng lots of nterest n geometrc group theory. The quas-sometrc rgdty phenomenon s looked for for many classes of spaces and groups. We frst recall the notons of quas-sometry and quas-sometrc embeddng whch s used to study coarse geometry. Defnton 1.1. Let (X, d X ) and (Y, d Y ) be metrc spaces. Gven real numbers L 1 and C 0,a map f : X Y s called a (L, C)-quas-sometry f (1) 1 L d X(x 1, x 2 ) C d Y (f(x 1 ), f(x 2 )) Ld X (x 1, x 2 ) + C for all x 1 and x 2 n X, and, (2) the C neghborhood of f(x) s all of Y. If f satsfes (1) but not (2), then f s called a (L, C)-quas-sometrc embeddng. Whle the set of quas-sometry rgdty results s qute rch by now, but not many results are known for quas-sometrc embeddngs. The rgdty of quas-sometrc embeddng generalzng quas-sometry rgdty would provde a parallel pcture of Marguls superrgdty generalzng Mostow rgdty. In ths paper, contnung from [FW18] and [FN], we study quas-sometrc embeddngs between hgher rank symmetrc spaces and lattces. The standng assumpton n ths paper s that X and Y are symmetrc spaces, Eucldean buldngs or lattces of the same R-rank and the R-rank s at least two. We also assume that Eucldean buldngs are thck. We study quas-sometrc embeddngs from X nto Y. We show that studyng embeddngs n ths general settng can be reduced to studyng embeddngs between symmetrc spaces and Eucldean buldngs where the target space s rreducble. That means we are able to groups rreducble factors of X nto new factors wth the same rank as rreducble factor of Y and the embeddng decomposes as a product of embeddng embeddngs from new factors of X nto rreducble factors of Y. In partcular we have the followng theorem. Date: October 22, Key words and phrases. quas-sometrc embeddng, rgdty, symmetrc space, Eucldean buldng, lattce. 1

2 2 THANG NGUYEN Theorem 1.2. Let X = X 1 X n and Y = Y 1 Y m be symmetrc spaces or Eucldean buldngs wthout compact and Eucldean factors, where X, Y j are rreducble factors. Let f : X Y be a (L, C)-quas-sometrc embeddng. Then there s a decomposton X = X 1... X m and maps (f 1,..., f m ) where (1) X s a product of rreducble factor of X and rank R(X ) = rank R(Y ), for = 1,..., m.. (2) f : X Y s a quas-sometrc embeddng for = 1,..., m. (3) f s at a bounded dstance from (f 1,..., f m ). By usng asymptotc cone technque, Theorem 1.2 s reduced to a results on b- Lpschtz embeddngs on Eucldean buldngs. To state our result n that language, we frst need to recall some termnology. In each flat of an Eucldean buldng, each Weyl subflat are parallel wth the annhlator of a fnte unon of roots. Each fnte unon of roots determnes the type of the subflats. We have the followng defnton. Defnton 1.3. A R-branched Eucldean buldng s an Eucldean buldng n whch unon of Weyl hyperplanes of each type s the whole buldng. Smplcal trees and standard Eucldean buldngs are not R-branched buldngs. Ther asymptotc cones and also asymptotc cone of symmetrc spaces are examples of R-branched Eucldean buldngs. In an R-branched buldng, each pont n a flat s contaned n Weyl hyperplanes. The man reducton of Theorem 1.2 s the followng theorem. Theorem 1.4. Let X = X 1 X n and Y = Y 1 Y m be R-branched Eucldean buldngs wthout Eucldean factors, where X and Y j are rreducble factors for 1 n and 1 j m. Let f : X Y be a b-lpschtz embeddng. Then there s a decomposton X = X 1... X m and f = (f 1,..., f m ) where (1) X s a product of rreducble factors of X and rank R(X ) = rank R(Y ), for = 1,..., m.. (2) f : X Y s a b-lpschtz embeddng for = 1,..., m. We remark that a factor f could be chosen to be sometrc f certan assumpton on types of X and Y s satsfed. However, due to the exstence of quas-sometrc embeddngs that are not close to sometrc embeddngs (see AN-map n [FW18, Proposton 2.1], and the exstence of rank one factors, we can always fnd an example where not all factors f of f s close to homothetc embeddng. Combng Theorem 1.2 wth results from [FW18] We can get a rgdty of the quas-sometrc embeddng n the followng stuaton. Corollary 1.5. Let X, Y be symmetrc spaces or Eucldean buldng wthout compact and Eucldean factors. Assume rank R (X) = rank R (Y ). Furthermore assume the collectons of type A patterns for X and Y are the same (wth multplcty). Let f : X Y be a quas-sometrc embeddng. Then f s at a bounded dstance from a product of sometrc embeddngs, up to rescalngs, of hgher rank factors and quas-sometrc embeddngs of rank one factors. We also have a rgdty phenomenon for rreducble non-unform lattces. Frst we show that quas-sometrc embeddngs of an rreducble non-unform lattce n a semsmple but not smple Le group can be reduced to the study of quas-sometrc embeddng of symmetrc spaces nto an rreducble one.

3 QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES AND LATTICES: REDUCIBLE CASE3 Theorem 1.6. Let Γ be nonunform lattces wth trval center n hgher rank semsmple Le group G, G be a semsmple Le group of the same rank as G and the rank at least 2. Assume that G = G 1 G n, G = G 1 G m where G, G j are connected smple Le group. We also assume that G and G have no Eucldean or compact factors and have the same collecton of type A s, wth multplcty. Furthermore we assume that G s not smple,.e. m 2. Let ϕ : Γ G s a QIembeddng. Then there exsts a decomposton G = G 1 G m and embeddngs f : G G such that (1) rank R (G ) = rank R(G ). (2) f are quas-sometrc embeddngs, for all 1 m. (3) f = (f 1,..., f m ) Γ s at a bounded dstance from ϕ. Furthermore, f s an sometrc embeddng, up to rescalng, f rank R (G ) > 1. Ths helps us to obtan a rgdty for embeddngs of rreducble non-unform lattces n sem-smple groups. Theorem 1.7. Let Γ, Λ be nonunform lattces n hgher rank semsmple Le groups G, G of the same rank and rank at least 2. Assume centers are trval and G, G have no compact or rank one factor. Furthermore, assume: (1) G and G have the same collecton of type A (wth multplcty), and (2) there s no closed subgroup G < H < G wth compact H orbt on Λ\G. If ϕ : Γ Λ s a QI-embeddng, then ϕ s at bounded dstance from a homomorphsm Γ Λ where Γ < Γ has fnte ndex. Fnally, we want to remark that quas-sometrc embeddng s much more flexble than quas-sometry n general. Outsde of rgd stuaton, not much s known about quas-sometrc embeddngs. We present some non-rgd quas-sometrc embeddng examples. Theorem 1.8. We have followng examples (1) There s a quas-sometrc embeddng from a product of two trees nto an A 2 -buldng such that there s no flat maps nto a neghborhood of a flat. (2) There s a quas-sometrc embeddng H 2 H 2 SL(3, C)/SU(3) such that there s no flat maps nto a neghborhood of a flat. Our constructon does not gve examples about non-rgd embeddngs between rreducble spaces. Thus t s natural to ask what an embeddng between rreducble spaces looks lke. The only know embeddngs known to the author constructed n [FW18], are compostons of AN-maps. We have the followng questons, see also [FW18, Secton 5]. Queston 1.9. Does there exst a quas-sometrc embeddng whch s not a composton of AN-maps? Let us brefly menton the hstory of quas-sometrc rgdty of symmetrc spaces and lattces. One of the frst and motvatng result s Mostow s rgdty [Mos68], whch can be formulated n terms of quas-sometry: a quas-sometry of real hyperbolc spaces of dmenson at least 3 or of hgher rank symmetrc spaces that s equvarant under an somorphsm of unform lattces s close to an sometry. Prasad extends the result to nonunform lattce n [Pra73]. The rgdty of other rank one symmetrc spaces s obtaned by Pansu s [Pan89]. For hgher rank symmetrc spaces

4 4 THANG NGUYEN and buldngs, Klener-Leeb and Eskn-Farb show that quas-sometres are close to sometry or homothety n [KL97] and [EF97]. Ther quas-flat result has been one of the man tools n study quas-sometry of groups actng geometrcally on CAT(0) spaces. For non-unform lattces, quas-sometry s even more rgd: frst was obtaned n a strkng result [Sch95] of Schwartz for rank one, and later n Schwartz s [Sch96], Farb-Schwartz [FS96], Eskn s [Esk98], Drutu s [Dru00] for hgher rank Le groups. Quas-sometrc embeddngs of hgher rank rreducble symmetrc spaces was frst studed by Fsher-Whyte [FW18]. In a work of the author wth Davd Fsher [FN], we studed embeddngs non-unform lattces n smple Le groups. Let us menton how the paper s organzed. In Secton 2, we study somorphc lnear maps between Eucldean vector spaces that preserve Weyl pattern. The key fact we obtan from ths secton s that such lnear map send each rreducble factor nto an rreducble factor. Secton 3 extends the clam rreducble factor maps to an rreducble factor to hold true for b-lpschtz embeddngs between R- branched Eucldean buldngs. We prove Theorem 3.7, whch s the key step to obtan further rgdty results. The argument s a combnaton of dfferentaton, results from Secton 2, and combnatoral argument at non-dfferentblty ponts. The rgdty of quas-sometrc embeddng rgdty between symmetrc spaces and buldngs, Theorem 1.2 and Corollary 1.5 when there are addtonal assumpton on Weyl patterns also follows. Next secton, we study quas-sometrc embeddng of non-unform lattces. And n the last secton, we nclude a dscusson of some non-rgd embeddngs, n partcular Theorem 1.8. Acknowledgment: I would lke to thank Davd Fsher for ntroducng the questons together wth very helpful dscussons, and explanaton about argument n [FW18]. Part of the work was done durng the author s vst at Wezmann Insttute. We would lke to thank Ur Bader and Wezmann Insttute for the hosptalty. Fnally, we want to thank Robert Young for useful dscussons on non-rgd embeddng examples and quas-sometrc embeddng n general. 2. Weyl pattern preservng lnear maps In ths secton, we study lnear maps that preserve Weyl patterns [FW18, Defnton 4.1]. Those lnear maps wll be used for studyng dervatves of blpschtz maps restrcted to flats n asymptotc cones. The man result of ths secton s the followng theorem. Theorem 2.1. Any lnear map that preserves Weyl patterns, maps each rreducble factor nto an rreducble factor. The dea for ths theorem s that f the lnear map does not send rreducble subflat nto one rreducble factor then the mage splts nto a product. On the other hand, the rreducble pattern ntutvely s always more complcated than the reducble pattern,.e product pattern, of the same rank. Ths makes such lnear embeddng s mpossble. To make a rgorous proof, we argue by nducton on dmenson of the doman. We frst ntroduce some notons and lemmas. Defnton 2.2 (Restrcted pattern). Let V be a vector space wth Weyl pattern, and let W be a sngular subspace. The restrcted pattern for W s the pattern n whch sngular subspaces of the restrcted pattern are ntersectons of W wth sngular subspaces of V.

5 QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES AND LATTICES: REDUCIBLE CASE5 Remark 2.3. Restrcted pattern may not be lnearly equvalent wth a Weyl pattern of the approprate dmenson. For example, let V = R 4 wth a Weyl pattern of type D 4. We assume that the Weyl hyperplanes are {x = ±x j : 1 < j 4}. The restrcted pattern to the hyperplane x 1 = x 2 has 7 restrcted hyperplanes whch are x 1 = x 2 = 0, x 1 = x 2 = ±x 3, x 1 = x 2 = ±x 4, and x 1 = x 2 x 3 = ±x 4. Therefore ths restrcted pattern s not lnear equvalent wth any Weyl pattern of dmenson 3. Before provng the theorem, we nvestgate possble restrcted patterns. We study ths for type A patterns and type D patterns. General patterns wll follow from behavor of restrcted pattern n type D. Frstly, n type A pattern, we have the followng lemma. Lemma 2.4. Any sngular subspace of an A n -pattern vector space wth the restrcted Weyl pattern can be lnearly dentfed wth a type A Weyl pattern vector space of dmenson of the subspace. Proof. It suffces to prove the lemma for co-dmenson one sngular subspaces. Pck a presentaton of A n vector space to be and Weyl hyperplanes are W = {(x 1,..., x n+1 ) R n+1 : x x n+1 = 0}. W j = {(x 1,..., x n+1 ) V : x = x j }, for 1 < j n + 1. It s enough to show that W 12 wth restrcted Weyl pattern can be lnearly dentfed wth a vector space of type A n 1. Indeed, Weyl hyperplanes ntersect W 12 and form followng restrcted hyperplanes n W 12 : for all 3 < j n + 1, and W j = {(x 1,..., x n+1 ) W 12 : x 1 = x 2, x = x j }, W 12k = {(x 1,..., x n+1 ) W 12 : x 1 = x 2 = x k }, for 3 k n + 1. Let U = {(y 1,..., y n ) : y 1 + +y n = 0} be a vector space of A n 1 type, n whch Weyl hyperplanes are U j = {y = y j } U, for 1 < j n. Then the lnear map f : U W 12, defned by f(y 1,..., y n ) = (y 1, y 1, n+1 n y 2 1 n y 1,..., n+1 n y n 1 n y 1) maps Weyl hyperplanes U j to W (+1)(j+1), for 1 < < j < n, and U 1k to W 12(k+1), for 1 < k n. Ths lnear map dentfes W 12 together the restrcted Weyl pattern wth a vector space of A n pattern. Thus the lemma follows. We now study restrcted pattern n type D vector spaces. Type D pattern s the smplest pattern among patterns that s not of type A. It s smplest n the sense that other patterns contan type D as a sub-pattern. The result for other types wll follow as a corollary. Lemma 2.5. Let V be a D n vector space, n 3. There s a chan of sngular subspaces V 2 V 3 V n 1 V such that dm(v ) = and V wth restrcted pattern contans a lnearly embedded type D pattern. Moreover, the number of of restrcted hyperplanes n V s strctly greater than the number of hyperplanes n a type D vector space, for 2 n 1.

6 6 THANG NGUYEN Proof. Assume that V s equpped wth the canoncal type D n Weyl hyperplanes V + j = {(x 1,..., x n ) : x = x j }, and V j = {(x 1,..., x j ) : x = x j }, for 1 < j n. It suffces to show that there s a sngular hyperplane wth restrcted pattern contanng D n 1 pattern. We show that V 12 + satsfes ths clam. Indeed, restrcted hyperplanes of V 12 + are V 12 = {(0, 0, x 3,..., x n )} and V ± j = {(x 2, x 2, x 3,..., x n ) : x = ±x j } for 2 < j n. The lnear map f : R n 1 W 12 +, defned by f(y 1,..., y n ) = (y 1, y 1, y 2,..., y n ), maps R n 1 wth canoncal D n 1 pattern nto V 12 + where the maged pattern conssts of all V ± j, for 2 < j n. We also see that the restrcted hyperplane V 12 s not n the collecton of maged hyperplanes of R n 1 by f. We make the followng defnton. Defnton 2.6. Let V be a vector space of n dmensons. A chan of vector subspaces V 2 V 3 V n 1 V n s called successve f dm(v ) =, for 2 n. We have the followng corollary from Lemma 2.5. Corollary 2.7. Let V be a vector space of one of the followng types: E 6, E 7, E 8, F 4, and BC n, D n for n 3. Then V contans a successve chan of sngular subflats such that each subflat wth restrcted pattern contans a lnearly embedded type D pattern of the same dmenson. Moreover, the number of of restrcted hyperplanes n each sngular subflat s greater than the number of hyperplanes n a type D vector space of the same dmenson. Proof. If V s of one of the lsted types then V contans a lnearly embedded type D n pattern. The corollary now follows by applyng the Lemma 2.5 repeatedly. We conclude the secton by a proof of the man theorem of ths secton. Proof of theorem 2.1. Let T : V 1 V 2 be a lnear map that preserves Weyl patterns. It suffces to prove the theorem n the case the pattern of V 1 s rreducble,.e. we show that T maps V 1 nto an rreducble Weyl factor of V 2. We know that sngular subspaces map to sngular subspaces. And sngular subspaces n the target are products of sngular subspaces n the rreducble factors. The clam s trval of rank f V 1 s 1. Thus, we can assume that rank of V 1 s at least 2. Frst, we note that the clam s true when V 1 s of type G 2. Snce the only reducble rank 2 vector space s of type A 1 A 1, whch has only two hyperplanes whle a type G 2 vector space has 6 hyperplanes. Thus mage of a G 2 plane cannot be reducble. Suppose that T does not send V 1 nto one rreducble factor, then there exst sngular subspaces W 1 W 2 V 1, such that dm(w 1 ) = dm(w 2 ) 1 and T maps W 1 nto an rreducble factor but does not map W 2 nto an rreducble factor. We then have T (W 2 ) = T (W 1 ) R and the restrcted hyperplanes of T (W 2 ) are exactly ether T (W 1 ) or products of restrcted hyperplanes of T (W 1 ) and the R factor. In partcular, the ntersecton of all but one hyperplanes n T (W 2 ) s a 1 dmenson sngular subspace. Case 1: V 1 s of type A n. Then restrcted patterns of W 1 and W 2 are (lnearly dentfed) of type A. However, n type A, ntersecton of all but one hyperplanes s always trval ({0}). Ths contradcts wth the phenomenon of the mage T (W 2 ) as we remark above. Case 2: V 1 s of one of the types BC n, D n, E 6, E 7, E 8, F 4. Suppose frst that dm(w 1 ) 2. In ths case we can choose W 1, W 2 to be from a successve chan

7 QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES AND LATTICES: REDUCIBLE CASE7 of sngular subspaces that are obtaned from Corollary 2.7. Restrcted pattern of W 1, W 2 contans embedded type D patterns. In type D vector spaces of dmenson at least 3, ntersecton of all but one hyperplanes s trval. Hence, we get the same contradcton as n type A case. Therefore, the only possblty left s when dm(w 1 ) = 1 and dm(w 2 ) = 2. In ths case, T (W 2 ) has two restrcted hyperplanes. On the other hand, W 2 can be chosen to be the dmenson 2 subspace n the chan obtaned Corollary 2.7. It follows that W 2 has at least 3 restrcted hyperplanes. Thus, T maps W 2 wth at least 3 restrcted hyperplanes nto T (W 2 ) havng only 2 restrcted hyperplanes. Ths contradcts wth assumpton that T preserves pattern. Therefore, any pattern preservng lnear map maps each rreducble factor nto an rreducble factor. 3. Decomposton of quas-sometrc embeddngs Snce D 2 = A 1 A 1 and D 3 = A 3, we only consder them as type A patterns to avod confuson. And from now on, we have the type D n for only n 4. In ths secton, except n proof of Theorem 1.2 and Corollary 1.5 at the end of the secton, we assume that X = X 1 X n and Y = Y 1 Y m are R-branched Eucldean buldngs wthout Eucldean factors, where X, Y j are rreducble factors. We let f : X Y be a b-lpschtz embeddng. Defnton 3.1 (Irreducble Subflat). An rreducble subflat s a subflat that satsfes followng condtons () the subflat s contaned entrely n an rreducble factor,.e. the projecton to the product of other factors s a pont, and () the subflat has the same rank as the rank of the rreducble factor. By [KL97, Theorem 7.2.1, Corollary 7.2.4], each b-lpschtz flat s contaned n a fnte unon of flats. Moreover, we could deduce the followng fact. Lemma 3.2. Each b-lpschtz flat s a fnte unon of convex polyhedra. We note that there s a coarse verson of ths lemma n Eskn s [Esk98, Lemma A.3]. The dea s the same but the argument for b-lpschtz flats n asymptotc cones s much smpler, based on Klener-Leeb s [KL97]. We gve a proof of ths lemma for completeness. Proof. Every b-lpschtz flat s contaned n a fnte unon of flats. The unon of flats can be wrtten as a fnte unon of closed convex polyhedra. The b-lpschtz flat s closed, hence so s ts ntersecton wth a polyhedron. If a polyhedron n the unon of polyhedra contans an nteror pont belongng to the b-lpschtz flat, then the whole polyhedron s contaned n the b-lpschtz flat. Indeed, suppose that there s an nteror pont x belongng to the b-lpschtz flat but any ts neghborhood does not. Locally at x, the b-lpschtz flat s a cone over a sphere whch s a fnte unon of sectors [KL97, Corollary 6.2.3]. Snce x s an nteror pont of the polyhedron, those Weyl cones s completely n the polyhedron. Thus, there s a neghborhood of x beng contaned completely n the polyhedron. Corollary 3.3. Image of a k-sngular subflat under a b-lpschtz embeddng of a Eucldean buldng s contaned n a fnte unon of k-sngular subflats.

8 8 THANG NGUYEN Proof. Consder two flats whose ntersecton s the gven k-sngular subflat. Image of each flat s a fnte unon of convex polyhedra. Image of the k-sngular subflat s the ntersecton of the two fnte unons. B-Lpschtz maps preserve dmenson. Hence mage of a k-sngular subflat s contaned n a fnte unon of k-sngular subflats. We now can deduce the local behavor of sngular subflats n the followng lemma. We note that ths lemma can also be found n [KL97, Corollary 6.2.3], though t s stated only for maxmal dmenson flats there. Lemma 3.4. Let F be a flat n X, M be a k-dmensonal sngular subflat of F, and x be a pont n M. Then there s a neghborhood of x n M such that the mage of that neghborhood s a cone wth vertex f(x) over a sphere of dmenson (k 1). Proof. Let F 1 be another flat such that M = F F. By [KL97, Corollary 6.2.3], there are neghborhoods U F, U F of x such that f(u) and f(u ) are cones over spheres wth vertex at x. Locally at x, the mage of M s contaned n f(u) f(u ). The ntersecton s locally a k-dmensonal cone. Locally at x, the mage of M s a k-dmensonal dsc. Thus the concluson follows. For x X, y Y, we denote Σ x X and Σ y Y tangent cones of X and Y at x and y. We have that Σ x X and Σ y Y are sphercal buldngs. If f : X Y blpschtz, f(x) = y then f nduces a map f = Σ x X Σ y Y whch s a homeomorphsm nto the mage. Sngular subflats at x correspond to walls n Σ x X. The nduced map sends each wall nto unon of walls. We defne a followng termnology. Defnton 3.5. A subset s called beng contaned entrely n one (rreducble) factor f projectons of the subflat to all but one factor are just ponts. We also say that a subflat maps nto a factor f ts mage s contaned entrely n a factor. Let us brefly menton how to show that b-lpschtz embeddngs decompose nto product. Frst, we show each rreducble subflat maps entrely n a factor. Next, we show the restrctons of embeddngs on flats decompose a product map. And fnally we promote the decomposton of restrctons on each flat to the decomposton of the embeddngs. The frst step, provng every rreducble subflat maps entrely n a factor for rank one subflats and hgher rank subflats requre dfferent treatments. We have the followng lemma for rank one rreducble subflats. Lemma 3.6. Let f : X Y be a b-lpschtz embeddng between real R-branched Eucldean buldngs of the same rank. Then mage of each rreducble rank one subflat s contaned entrely n one rreducble factor of Y. Moreover, mage of two rreducble rank one subflats of the same factor are contaned n the same rreducble factor of Y. Proof. Frst, we note that the number of rank one factors of X s not smaller than the number of rank one factors of Y. Indeed, by the same argument as n [FW18, FN], n each flat, ponts of dfferentablty are generc. At those ponts, dervatves preserve Weyl pattern. By Theorem 2.1, dervatves map each rreducble factor nto an rreducble factor. If an rreducble factor of X has rank at least two then t can not map lnearly and njectvely nto a rank one factor of Y. Thus the number of rank one factors of X cannot be smaller than the number of rank one factors of Y.

9 QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES AND LATTICES: REDUCIBLE CASE9 Let F be an arbtrary flat n X. By [FW18, Lemma 3.1], outsde of a codmenson two subset S of F, f locally maps F nto a flat. We frst show that f maps each rreducble rank one subflat of F that s dsjont from S entrely nto a factor. Let F 1 be such an rreducble rank one subflat. We have that f(f 1 ) s a unon of fntely many geodesc (possbly nfnte) segments n rank one subflats. Each rank one subflat must be contaned n a factor. Thus t suffces to show that two consecutve segments belong to a same factor. Let z be a common end pont of the two consecutve segments. Snce z s not n S, there s a flat F contanng f(z) such that f locally maps F at z nto F at f(z). Thus f nduces a homeomorphsm Σ z X Σ z F Σ f(z) Y Σ f(z) F, where Σ z F and Σ f(z) F are tangent cones of F and F at z and f(z) respectvely. We denote F1 the co-dmenson one sngular subflat of F at z. Ths co-dmenson one subflat s also the orthogonal complement of F 1 n F. Then we have that Σ f(z) f(f 1 ) Σ f(z) f(f1 ) =. The fact f(f1 ) s a unon of co-dmenson subflats mples that Σ f(z) f(f1 ) s a unon of co-dmenson one walls n Σ f(z) F. Snce Σ z F1 separates Σ x F 1 n Σ x F, we also have Σ f(z) f(f1 ) separates Σ f(z) f(f 1 ). Now suppose that f maps F 1 nto two dfferent factors at z then Σ f(z) f(f 1 ) s a unon of two ponts whch s not n a jont factor of Σ f(z) F. It follows that these two ponts are connected by an arc of length π 2. Ths arc does not contan any sngular pont n ts nteror, thus cannot be separated by a unon of walls. Ths s contradcton. Thus f(f 1 ) does not change factors at f(z). Therefore, f(f 1 ) s contaned entrely n factor. We now show that f maps all rreducble rank one subflats n F that are parallel wth F 1 entrely nto a factor. Every rreducble rank one subflat n F that s dsjont from S maps entrely to a factor by above argument. If moreover the subflat s parallel wth F 1 then t maps tho the same factor as F 1 maps to because of fnte Hausdorff dstance between two subflats. Let F 1 be an arbtrary subflat parallel wth F 1 n F. Snce S s of co-dmenson two, F 1 can be approxmated by a sequence subflats that are dsjont from S. Images of the sequence of approxmate subflats have projectons to all but one factors are just ponts. It follows that f(f 1 ) has projectons to all but one factors are just ponts,.e. f(f 1 ) s contaned entrely n a factor. Therefore, f maps each rreducble rank one subflat n F entrely nto one rreducble factor of Y. The clam for general rreducble rank one subflats n X follows from the fact that gven two arbtrary flats n X, there s a fnte sequence flats n X wth two gven flats beng the frst and last flats n the sequence and two consecutve flats n the sequence ntersect n a half-flat. We now can decompose a b-lpschtz embeddng nto product of embeddngs nto rreducble factors by the followng theorem. The clam that each rreducble hgher rank subflats maps entrely nto a factor s also contaned n proof of the theorem. Theorem 3.7. Let X = X 1 X n and Y = Y 1 Y m be R-branched Eucldean buldngs wthout Eucldean factors, where X, Y j are rreducble factors. Let f : X Y be a b-lpschtz embeddng. Then there s a decomposton X = X 1... X m and f = (f 1,..., f m ) where (1) X s a product of rreducble factors of X and rank R(X ) = rank R(Y ), for = 1,..., m. (2) f : X Y s a b-lpschtz embeddng for = 1,..., m.

10 10 THANG NGUYEN Proof. Let F be a flat. We wrte F = F 1 F n, a product of subflats n rreducble factors. By [FW18, Lemma 3.1], there s a set Σ F that s the complement of a co-dmenson two subset, such that on Σ, f locally maps F to a flat. Snce f s b-lpschtz hence s dfferentable a.e. on Σ. At a pont of dfferentablty, the dervatve Df s a lnear map, preservng Weyl pattern. By Theorem 2.1, Df maps each rreducble factor nto an rreducble factor. Thus f locally maps each rreducble subflat through a pont of dfferentablty nto an rreducble subflat. Consder an rreducble subflat of form F {u} F, such that almost every pont n F {u} are n Σ and f s dfferentable a.e. on F {u}. By Lemma 3.6, f rank of F s one, then f(f {u}) s contaned entrely n one rreducble factor. We show that ths s also true n the case the rank of F s at least two. Suppose that the rank of F s at least two. We have f(f {u}) s contaned n a unon of fntely many subflats of the same dmenson. Locally, at ponts of dfferentablty, f maps F {u} nto a factor of Y. Snce n F {u}, the set of ponts of dfferentablty of f F s dense, and at those ponts the subflats are contaned n rreducble factors, we conclude that f(f {u}) s contaned n a fnte unon of subflats n whch each subflat n the unon s contaned entrely n a factor. Now consder an arbtrary pont (w, u) F {u}. By Lemma 3.4, there s a neghborhood of (w, u) n F {u} such that the mage of the neghborhood s a cone over a sphere. Because f(f {u}) s contaned n a fnte unon of subflats n whch each subflat s contaned entrely n a factor, ths sphere s contaned n the fnte unon of rreducble subflats. Because rank of F s at least 2, ths sphere s connected. Thus, t s contaned entrely one rreducble factor. It follows that the neghborhood of (w, u) maps nto one rreducble factor. Therefore, locally at any pont, F {u} maps nto one factor. Connectedness mples that the whole subflat F {u} maps nto one factor. Let F {u 1 } and F {u 2 } be two parallel subflats such that almost every pont n F {u 1 } F {u 2 } s n Σ and f s dfferentable a.e. on F {u 1 } F {u 2 }. Then each of the subflats F {u 1 } and F {u 2 } maps nto one factor. Snce two subflats have fnte Hausdorff dstance from each other, they map nto the same factor. Furthermore, any subflat F {u} can be approxmated by a sequence of parallel subflats F {u n }, where a.e. pont n F {u n } s not n Σ and s a pont of dfferentablty. Every F {u n } maps nto one rreducble factor, for all n. Hence F {u} maps nto the same factor as F {u n } does. We decompose, and possbly re-arrange, X = X 1 X m and Y = Y 1 Y m such that for all 1 m, Y s rreducble, X = X 1 X k() s a product of rreducble factors, and rank R (X ) = rank R(Y ), If we wrte F = F 1 F m, where F X then f maps the sngular subflat F nto the factor Y. We now show that every X -subflat maps nto the factor Y. Let F be an X - subflat such that F ntersects F n a half-subflat. We then have that the flat F = F 1 F 1 F F F m ntersects F n a half-flat. We know that all parallel X -factor subflats of F map nto a factor. The same s true for parallel X -factor subflats of F. Snce there are X -factor subflat of F and X -factor subflat of F ntersectng at a half-subflat, t follows easly that X -factor subflats of F and X -factor subflats of F map nto the same factor. Note that any two flats n a

11 QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES AND LATTICES: REDUCIBLE CASE 11 buldng can be connected by a sequence of flats n whch any two consecutve flats ntersectng n half-flats. Therefore, we conclude that all X -factor subflats of X map nto the same factor. As X -subflats of F maps nto the factor Y, we have that every X -subflat n X maps nto the factor Y. Next, we show that the map f can be decomposed as a product map. Because X -subflats map nto the factor Y, we have that for x X ( = 1,..., m) there exst y j (x 1,..., x 1, F, x +1,..., x m ) Y j for j such that proj Yj (f({x 1 } {x 1 } F {x +1} {x m })) = y j. On the other hand f F s a flat n X whch has nonempty ntersecton wth F then proj Yj (f({x 1 } {x 1 } F {x +1} {x m })) = y j. Ths s because the projecton of the mage of a X -subflat to Y j factor s a pont, and thus projectons of mages of two X -subflats wth nonempty ntersecton on Y j factor concde. It follows that y j does not depend on the X -coordnate. In other words, y j depends only on X j -coordnate. Therefore, f can be decomposed as a product map,.e. there are map f : X Y for = 1,..., m such that f = (f 1,..., f m ). In concluson, after possbly re-arrangng and re-ndexng, we decompose X = X 1 X m and f = (f 1,..., f m ) such that f : X Y s a b-lpschtz embeddng. We are now able to decompose general quas-sometrc embeddngs between symmetrc spaces and Eucldean buldngs n general nto product of embeddngs nto rreducble targets. The dea s usng asymptotc cone argument, we can transfer the conclusons for b-lpschtz embeddngs between R-branched buldngs to symmetrc space and Eucldean buldng settngs. Proof of Theorem 1.2. Frst we show the clam: there s a constant D(L, C, X, Y ) such that for any subflat of form F 1 {u}, there exsts 1 m so that dam(proj j (f(f 1 {u})) < D for all j. Suppose ths clam s not true, then there are rreducble X 1 -subflats S n such that projectons of mages of the subflats S n on at least two factors have dameters tendng to nfnty. We choose a sequence of numbers (c n ) so that c n s the mnmum of n and two dameters of projectons of f(s n ) to two factors. It follows that the nduced b-lpschtz map on asymptotc cones w.r.t. the sequence of rescalngs (c n ) and the sequence of centers on (S n ) does not map an rreducble subflat entrely nto an rreducble factor. Ths contradcts wth Theorem 3.7. Moreover, the argument apples to not only f but to all (L, C)-quas-sometrc embeddngs. Hence the constant D depends only on constants L, C and spaces X, Y. By the same argument as when we show the b-lpschtz map decomposes as a product f rreducble sunflats map nto rreducble factors n the proof of Theorem 3.7, after rearrangng and groupng rreducble factors, we can decompose X = X 1 X m such that projectons of mages of subflats of form F { } on factors Y j have fnte dameters for all j. Thus there are maps f : X Y for = 1,..., m such that f s at a bounded dstance from the product map (f 1,..., f m ). The fact f are quas-sometrc embeddngs for all 1 n follows from f X s a quas-sometrc embeddng and unformly close to f. Before provng the rgdty result n Corollary 1.5, we recall two mportant facts from classfcaton of Weyl pattern preservng lnear maps n [FW18, Secton 4]. The frst one s that there s no Weyl pattern preservng lnear map from non A- type nto an A-type target [FW18, Corollary 4.10]. And the second one s that a

12 12 THANG NGUYEN Weyl pattern preservng lnear map nto an rreducble space can be non-conformal only f the doman has A-type factor whle the target ether s not of A-type or s of A-type wth dfferent rank [FW18, Lemma 4.6, Lemma 4.7, Corollary 4.10]. Proof of Corollary 1.5. By Theorem 1.2, f splts as a product of quas-sometrc embeddngs. We need to show that each embeddng factor s at a bounded dstance from an sometrc embeddng, up to rescalng. f nduces a b-lpschtz map on asymptotc cones [f] : [X] [Y ]. We obtaned from theorem 1.2 a decomposton X = X 1 X m such that f : X Y s a quas-sometrc embeddng for 1 m. We look at asymptotc cones and the nduced map [f] = ([f 1 ],..., [f m ]). Consder one factor map. We note that f we have pattern preservng lnear map nto a type A pattern, then the doman must have at least an A-factor. Snce the collectons of A-types of the doman and the target are the same, the doman of a factor map nto a A-type pattern s of A-type wth the same rank. The assumpton X and Y havng the same collecton of type A pattern (wth multplcty) also mples that any pattern preservng lnear factor map from [X ] nto [Y ] s conformal. Ths s equvalent wth the clam that any lnear map preservng Weyl pattern of X nto Weyl pattern of Y s conformal. Then by [FW18, Theorem 1.8], f s at a bounded dstance from an sometrc embeddng, up to rescalng. 4. Embeddng of rreducble non-unform lattces In ths secton we study quas-sometrc embeddngs of non-unform lattces. We assume the settng of Theorem 1.6. Gven an embeddng of a non-unform lattce, we get an embeddng of a neghborhood of the lattce. However, the composton of ths map and nearest pont projecton does not gve a quas-sometrc embeddng of the whole symmetrc space. Thus we can not use drectly Corollary 1.5 from secton 3 and [FN, Theorem 1.4] to obtan a rgdty of the embeddng. Instead, we combne the deas and proofs n secton 3 and n [FN] paper to make an argument. The key pont s because of rreducblty of lattce, there are enough many flats and dvergng pattern to run argument. We quote here the results from [FN] whch do not need Le groups beng smple and refer to that paper for full detaled proof. Let ϕ : Γ Λ be a quas-sometrc embeddng of rreducble non-unform lattces. Then ϕ nduces a map X Y by pre-composng wth a closest pont projecton of X to Γ (pck one f there are more than one closest ponts) and post-composng wth the ncluson Λ Y. We note that an rreducble hgher rank lattce s quassometrc embedded nto the symmetrc space by [LMR00]. Abuse the notaton, we stll denote ϕ : X Y for the nduced map. Ths nduced map s not a quassometrc embeddng. However, an argument usng ergodcty shows that on many flats, the nduced map behaves almost lke a quas-sometrc embeddng. We assume that the semsmple groups satsfy the assumptons of Theorem 1.6. Let δ > 0. The followng proposton s obtaned from Fsher-Nguyen s [FN]. We note that the assumpton that Γ beng rreducble makes Howe-Moore ergodcty theorem can be appled. Proposton 4.1. [FN, Secton 3.1] Let ω be a non-prncpal ultraflter. There exsts a non-ncreasng functon θ δ convergng to 0 and a full measure famly F of sub-θ δ -dvergng flats n X such that for any sequence of flats F n that are subθ δ -dvergng w.r.t. x n, and any sequence c n wth lm c n = +, ϕ nduces a bω Lpschtz map [ϕ] [Fn]from [F n ] nto Cone(Y n, ϕ(x n ), c n, ω). The clam stll holds

13 QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES AND LATTICES: REDUCIBLE CASE 13 true for sequences of unons of fntely many ntersectng flats. Furthermore, for any unon of fntely many hyperplanes n [F n ], we could choose a unon of fntely many flats ntersectng [F n ] n the unon of hyperplanes. The defnton of a sub-θ δ -dvergng flat s qute techncal as t nvolves the use of ergodc theorem a few tmes. We refer to [FN, Secton 3.1] for a precse defnton. Roughly speakng, a flat s sub-θ δ -dvergng w.r.t. a fxed pont s a flat such that the proporton of measure of the subset of ponts n ball of radus r around the fxed pont of the dstance at least θ δ (r)r away from the lattce Γ goes to zero as r goes to nfnty. Proposton 4.2. For F n F as n Proposton 4.1, [ϕ] [Fn] s a product map nto Cone(Y, ϕ(x n ), c n, ω), and [ϕ] [Fn]([F n ]) s a flat. Moreover, each factor of [ϕ] [Fn], up to a rescalng, s a map gven by an element n Weyl group. Proof. The argument s a combnaton of the argument n [FN, Secton ] and the argument n Secton 3. To smplfy notaton, we denote [Y ] for Cone(Y, ϕ(x n ), c n, ω). We have [ϕ] [Fn]([F n ]) s a b-lpschtz flat thus s contaned n a unon of fntely many flats n [Y ]. By pckng other sequences of flats that have lmts ntersectng [F n ] n hyperplanes, we conclude that subflats of [F n ] map nto unons of fntely many subflats n [Y ]. In [F n ], off from a co-dmenson 2 subset, [ϕ] [Fn] locally maps flat to flat. Same argument as n Secton 3 mples that [ϕ] [Fn] s decomposed as a product map. Now, the doman and the range of each factor map satsfy the condton that all lnear map from the Weyl pattern of the doman to the Weyl pattern of the range s conformal. Hence each factor map sends flats to flats. It follows that [ϕ] [Fn]([F n ]) s a flat n [Y ]. Apply Proposton 4.2 to the sequence of a fxed flat F F, we get that there s a decomposton G = G 1 G m such that each G -subflat of [F ] maps nto G -subflat of [Y ]. We show that ths decomposton of G does not depend on the choce of F F. We note that f E F such that [F ] ntersects [E] n ether a halfflat, a Weyl chamber or a hyperplane, then the decompostons of G w.r.t. F and w.r.t E agree. As F has full measure n the set of flats, any two flats n F can be connected by a chan of flats n F whch any two consecutve flats coarsely ntersect n ether a half-flat, a Weyl chamber or a hyperplane. Therefore the decomposton s ndependent of the F F. Now we apply quantfy arguments n [FN] to obtan the followng proposton (whch s a combnaton of Proposton 3.3, Corollary 3.8, Proposton 3.9 and Proposton 4.1 n [FN]). We note that Γ s rreducble, hence all abelan subgroups correspondng wth subflats actng ergodc on Γ\G. Proposton 4.3. Let F be a sub-θ δ -dvergng flat w.r.t. x. Then there s a flat F Y such that all of the followngs hold true (1) ϕ(x) s at a bounded dstance from F and ϕ(f ) s sub-lnearly dvergent from F, (2) mages of Weyl chambers at x are sub-lnearly dvergent from a fnte unon of (a fxed number of) Weyl chambers n Y vertex at ϕ(x). (3) mages of a subflat n F contanng x s sublnear dvergent from a subflat n F contanng ϕ(x).

14 14 THANG NGUYEN (4) The set of y F such that F s a sub-θ δ -dvergng flat w.r.t. y has a large proporton of Lebesgue measure n F and the mages of such ponts are unformly close to F. Proof. See [FN, Proposton 3.3, Corollary 3.8, Proposton 3.9, Proposton 4.1]. Now we are ready to prove Theorem 1.6. Proof of Theorem 1.6. Let F F, consder F s the flat s obtaned by applyng Proposton 4.3. To smplfy the exposton, we abuse notatons and wrte F = ϕ(f ). We decompose G = G 1 G m such that for any F F, [ϕ] [F ] maps each G - subflat of [F ] to a G -subflat of [ϕ(f )]. Let X, Y be symmetrc spaces assocated to G and G. If x F s a pont such that F and all F α for α Ξ are sub-θ δ -dvergng w.r.t. x and let F = ϕ(f ), F α = ϕ[f α ]. We recall that the map ϕ now s the composton of orgnal ϕ and nearest pont projecton. If F F s sub-θ δ -dvergng w.r.t. x then x s R-close (R depends on Γ, G, and δ) to Γ, hence ϕ(x) s well defned up to some (2LR + C)-dstance,.e. ndependent of dfferent choces of pckng closest ponts n the projecton. We also recall that there s a subset Ω δ Γ\G whch s contaned n a R-neghborhood of p(1) Γ\G, such that a flat F F s sub-θ δ -dvergng w.r.t. x F f we can wrte x = π(g) and F = π(ga) for some g ΓΩ δ (See [FN, Secton 3]). We have the followng lemma. Lemma 4.4. If g, g Γ Nbhd 1 (Ω δ ) such that d(proj G (g), proj G (g )) < 1 then d(proj G (ϕ(g)), proj G (ϕ(g ))) < 12LR + 4D + 4L + 3C. We wll prove ths lemma after fnshng the proof of Theorem 1.6. We note that proj G (Γ Nbhd 1(Ω δ )) = G for all 1 m because Γ s rreducble. We defne a map ϕ : G G as follow: for every g G, we pck g Γ Nbhd 1(Ω δ ) such that proj G (g) = g. We defne f (g ) = proj G (ϕ(g)). By Lemma 4.4, ϕ s well-defned up a fnte dstance. We need to show that ϕ s a quas-sometrc embeddng from G nto G. Indeed, for g and g n G, by ergodcty of the acton of Ĝ = G 1 G 1 G G m on Γ\G, there are g and g n ΓΩ δ such that d(proj G (g), g ) < 1, d(proj G (g ), g ) < 1 and d(projĝ (g), projĝ (g )) < 1. Hence d(g, g ) d(g, g ) < 2. On the other hand, there s a flat F F ntersectng wth R-neghborhoods of g and g n ΓΩ δ. To smplfy notatons, let us denote C = 4(12LR + 4D + 4L + 3C). By trangle nequalty and ϕ beng quas-sometrc embeddng, we have It follows that d(projĝ (ϕ(g)), projĝ (ϕ(g ))) < C. d(ϕ(g), ϕ(g )) C < d(proj G (ϕ(g)), proj G (ϕ(g ))) = d(f (g ), f (g )) < d(ϕ(g), ϕ(g )). Snce ϕ s a quas-sometrc embeddng on a neghborhood of Γ, t follows that f s also a quas-sometrc embeddng. In the case that G has hgher rank, by [FW18] f s at a bounded dstance from an sometrc embeddng. In that case we can replace f by the sometrc embeddng.

15 QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES AND LATTICES: REDUCIBLE CASE 15 Fnally we show that the orgnal quas-sometrc embeddng s unformly close the the product map (f 1,..., f m ). Let γ = (γ 1,..., γ m ) Γ, there s g = (g 1,..., g m ) whch s n a R-neghborhood of γ n Ω δ. We have that L 1 R C < d(ϕ(γ), ϕ(g)) < LR + C, and L 1 R C C < d(f (g ), f (γ )) < LR + C + C. As ϕ and (f 1,..., f m ) are unformly close on ΓΩ δ, t follows that ϕ and (f 1,..., f m ) are unformly close on Γ. Now we prove Lemma 4.4. Proof of Lemma 4.4. Let γ, γ Γ such that g γ Nbhd 1 (Ω δ ) and g γ Nbhd 1 (Ω δ ). Snce the famly F has full measure n the set of flats, there s t G such that π(ta) s a flat n F and ta has non-empty ntersectons wth both γω δ and γ Ω δ. Let g ta γω δ and g ta γ Ω δ. Denote A be the G -factor abelan subgroup of A. The two subflats π( ga ) and π( g A ) are parallel n the flat π(ta). The Hausdorff dstance between the two subflats π( ga ) and π( g A ) s bounded by 2R. Snce Γ s rreducble, A acts ergodcally on Γ\G. By defnton of Ω δ, t follows that the sets ga and g A contan a large porton of ponts belongng to ΓΩ δ. Therefore, there are h ga j ΓΩ δ and h g A j ΓΩ δ such that d(h, h ) < 4R. It follows that d(ϕ(h), ϕ(h )) < 6LR + C. On the other hand, because g, g ΓΩ δ, there are G -subflats SF and SF n Y such that the mage under ϕ of ga ΓΩ δ and g A ΓΩ δ are D-close to SF and SF respectvely. We note that SF and SF are G -subflats, thus projecton of each of them to G factor s just a pont. Hence d(proj G (ϕ( g)), proj G (ϕ(h))) < 2D and d(proj G (ϕ( g )), proj G (ϕ(h ))) < 2D. In concluson, applyng the trangle nequalty we obtan d(proj G (ϕ(g)), proj G (ϕ(g ))) < 12LR + 4D + 4L + 3C. Proof of Theorem 1.7. When G s smple, the theorem s proved n [FN, Theorem 1.4]. We assume that G s not smple. By Theorem 1.6, ϕ s unformly close to a product of sometrc embeddngs. By [FN, Secton 6], we have that ϕ s unformly close the a group homomorphsm from a fnte ndex subgroup of Γ nto (possbly a conjugate by a commensurator of) Λ. Remark 4.5. Theorem 1.7 can also be proved by the general scheme: frst showng good flats map to flats, next usng [FN, Secton 5] to prove the regularty of boundary map and obtan a geometrc rgdty, and fnally usng [FN, Secton 6] to obtan the algebrac concluson. For the case G s not smple, the argument presented here s smpler as t passes around [FN, Secton 5], whch s the most dffcult part n argument for smple case. 5. Non-rgd quas-sometrc embeddngs 5.1. Combnatoral nterpretaton of AN-map. In ths subsecton we gve an way of nterpretng AN-map whch does not use any algebrac structure. Ths s helpful n constructng quas-sometrc embeddngs between Eucldean buldngs. We recall some defntons whch s used n [Lee00]: Let X be a symmetrc space or Eucldean buldng. X s a sphercal buldng. Gven a k dmensonal cell on X.

16 16 THANG NGUYEN Defnton 5.1 (Parallel Set and Cross Secton). [Lee00, Defnton 3.4] Gven a sngular unt sphere s X, the parallel set P (s) of s s the unon of flats or subflats whch have boundary at nfnty s s. The parallel set s sometrc wth a product of a Eucldean space and a symmetrc space or Eucldean buldng as P (s) = R dm(s)+1 CS(s). The symmetrc space or Eucldean buldng CS(s) s called the cross secton of s. We can also defne a cross secton of a smplcal cell n X as follow. Let s X be a sngular unt sphere. Let c s be a smplcal cell such that dm(c) = dm(s). We defne cross secton of c, denote CS(c), to be the cross secton of s. By [Lee00, Lemma 3.5], ths defnton s well-defned. Example 5.2. Let X be the A 2 -buldng assocated wth SL(3, Q p ). For any par of opposte sngular 0-cells ζ and ζ + on X then CS(ζ ) and CS({ζ, ζ + }) are homothetc to T p+1, where T p+1 s the (p+1)-regular tree. Moreover, cross sectons of all sngular ponts n X are sometrc. Example 5.3. Let X = SL(3, R)/ SO(3), a symmetrc space of type A 2. In X there are many totally geodesc copes of the hyperbolc plane H 2. Each copy of H 2 can be dentfed wth the cross secton of two boundary ponts of a sngular geodesc. We recall the defnton of spaces of strong asymptote classes from [Lee00, Secton 2.1.3]. For a pont ξ X we consder the set of geodesc rays asymptotc to ξ. The asymptotc dstance between two rays ρ 1, ρ 2 : [0, ) X s gven by d ξ (ρ 1, ρ 2 ) = nf d(ρ 1(t 1 ), ρ 2 (t 2 )). t 1,t 2 The space of strong asymptote classes assocated to ξ, denoted by X ξ s the space of geodesc rays asymptotc to ξ quotent out by equvalence relaton defned by zero d ξ -dstance. There s natural projecton proj ξ : X X ξ by mappng each x X to the asymptote class of the geodesc ray from x to ξ. We note that ths projecton s dstance non-ncreasng map. Moreover, f ξ s an nteror pont of k-dm cell c X then X ξ s sometrc to R k CS(c). In partcular, f ξ s a sngular 0-cell then X ξ and CS(ξ) are sometrc. Example 5.4. [FW18, Theorem 1.6] One of smplest examples of AN-map s a quas-sometrc embeddng f : H 2 H 2 SL(3, R)/SO(3). Frst we dentfy H 2 wth the set of uppertragular matrces n SL(2, R) and SL(3, R)/SO(3) wth the set of upper trangular matrces n SL(3, R). The map can be gven by an explct formula as follows f : (( ) ( )) e t xe t e s ze 0 e t, s 0 e s e 2 3 (t+s) xe 2 3 (s 2t) ze 2 3 (t 2s) 0 e 2 3 (s 2t) e 2 3 (t 2s) We call a flat vertcal f t s parametrzed as the set {(( ) ( )) } e t xe t e s ze 0 e t, s 0 e s : t, s R, for some fxed x, z R. We note that n ths example, each vertcal maps to a flat, n whch the vertcal quadrant and ts opposte map to unon of two Weyl sectors each. In partcular, there s an open and dense subset of Fursterberg boundary of

17 QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES AND LATTICES: REDUCIBLE CASE 17 H 2 H 2 such that f maps each chamber n the set to unon of two chambers n Fursternberg boundary of Y = SL(3, R)/SO(3). e 2 3 t 0 0 Let ξ 1 Y and ξ 2 Y be the endponts of geodesc ray 0 e 4 3 t 0 : t [0, ) 0 0 e 2 3 t e 2 3 s 0 0 and ray 0 e 2 3 s 0 : s [0, ) 0 0 e 4 3 s. It can be checked that proj ξ 1 f maps each copy of H 2 of from H 2 { } H 2 H 2 sometrcally to Y ξ1. Smlarly, proj ξ1 f maps each copy { } H 2 sometrcally to Y ξ2. On the other hand, f F Y s a flat such that ξ 1, ξ 2 F then every pont x n F s unquely determned by (proj ξ1 (x), proj ξ2 (x)). Moreover, every pont n the unon of flats contanng ξ 1 and ξ 2 on the boundares s unquely determned by projectons to Y ξ1 and Y ξ2. Indeed, let Y (ξ 1, ξ 2 ) be the set of unon of all flats contanng ξ 1 and ξ 2 on the boundares. Let x and z be two arbtrary ponts n Y (ξ 1, ξ 2 ). Two geodesc rays [x, ξ 1 ) and [x, ξ 2 ) bound a unque Eucldean sector E x whch s sometrc to unon of two Weyl sectors. Smlarly [z, ξ 1 ) and [z, ξ 2 ) bound a unque Eucldean sector E z. If (proj ξ1 (x), proj ξ2 (x)) = (proj ξ1 (z), proj ξ2 (z)) then E x = E z. It follows that x = z. Furthermore, (proj ξ1, proj ξ2 ) : Y (ξ 1, ξ 2 ) Y ξ1 Y ξ2 s a quassometry. And Y (ξ 1, ξ 2 ) s quas-sometrc embedded nto Y as Y (ξ 1, ξ 2 ) s a unon of vertcal flats. Now f can be obtaned by frst dentfy two factors H 2 wth Y ξ1 and Y ξ2 then compose t wth (proj ξ1, proj ξ2 ) 1. Example 5.5. We can now defne a smlar example of quas-sometrc embeddng T p+1 T p+1 Y, where Y s an A 2 Eucldean buldng assocated to SL(3, Q p ). Let ξ 1 and ξ 2 be two sngular ponts n Y that s of combnatoral dstance 2 apart. We have Y ξ1 and Y ξ2 are sometrc and are homothetc to T p+1. The map f = (proj ξ1, proj ξ2 ) 1 ) : T p+1 T p+1 Y s a quas-sometrc embeddng. Abuse the notaton, we call ths embeddng an AN-map. There s a natural queston that whether all quas-sometrc embeddng are of the form AN-map. The answer s negatve, due to the followng example. Example 5.6. [FW18, Theorem 1.7] There s an embeddng f from H 2 H 2 nto Sp(4, R), whch s not of the form AN-map. Indeed, we can obtan such an embeddng by composng two embeddngs of form AN-map. The AN-maps are H 2 H 2 SL(3, R) and SL(3, R) Sp(4, R). We recall that to defne an AN-map we have to fx a Weyl chamber at nfnty to defne the famly of vertcal flats. We choose the Weyl chamber n SL(3, R) defnng vertcal flat famly to be a Weyl chamber that s not n the mage of chambers under the embeddng H 2 H 2 SL(3, R). By examne numbers of Weyl sectors that each Weyl sector maps to under the composton t can be showed that f s not an AN-map. We remark that n above example there s flat mappng to a sngle flat. In fact, we can construct examples of quas-sometrc embeddngs whch s even more non-rgd: no flat maps to fnte neghborhood of a sngle flat. We have seen that showng flats map to flats s the frst key step to obtan rgdty of quas-sometres and quas-sometrc embeddngs. And an attempt of showng quas-sometrc embeddngs beng AN-map may have to start wth fndng a famly of flats that map