Lattices in semisimple groups and distal geometric structures. Five lectures on lattices in semisimple lie groups by. Let gbe a connected semisimple lie group without compact factors whose real rank is at least 2, and let. Twisted conjugacy classes in lattices in semisimple lie groups article pdf available in transformation groups 191 january 2012 with 63 reads how we measure reads. We will prove the main theorem for a special case when the discrete group in the main theorem is a fuchsian lattice. Isoperimetric inequalities for lattices in semisimple lie. Discrete subgroups and lattices in semisimple lie groups form a rich and wellstudied class of nitely generated groups acting on nonpositively curved metric spaces. It has been shown that both cocompact and noncocompact lattices exist inside every noncompact algebraic semisimple lie group. Rigidity for lattices in semisimple lie groups seminario matematico. Citeseerx document details isaac councill, lee giles, pradeep teregowda. One of the main successes of the theory of lattices is that it gave in a uni.
This paper is about semisimple lie groups and lattice subgroups therein. Now, if you have a noncompact semisimple lie group and a lattice in it, you can mod up the center and get an algebraic group and a lattice in it. If both a12 and a21 are equal to 0, then the product. See the history of this page for a list of all contributions to it. One quasiisometry class for each semsimple lie group g, consisting of the uniform lattices in g. This allows us to determine the subgroup growth of irreducible lattices of semisimple lie groups. The fundamental group of a compact irreducible manifold of nonpositive sectional curvature is either a lattice in a higher rank semisimple lie. Therefore they often serve as a test ground for general assertions on nonpositive curvature.
We characterize groups with nonlinear divergence functions as groups having cutpoints in their asymptotic cones. Lattices in semisimple lie groups, and multipliers of group. Harmonic analysis on semisimple lie groups harishchandra, bulletin of the american mathematical society, 1970. This gives a new proof of behrstocks result that every pseudoanosov element in a mapping class group is morse. Discrete subgroups isomorphic to lattices in semisimple lie groups. Lattices in semisimple groups and invariant geometric structures. Subgroup growth of lattices in semisimple lie groups core. As mentioned above, in the classical case of semisimple lie groups, the combination of the kazhdanmargulis theorem with local rigidity of lattices yields a much stronger niteness statement, due to wang wan72. Smooth classification of cartan actions of higher rank. A discrete subgroup t of is said to be a lattice in g if the. Stabilizers of lattices in lie groups heldermannverlag.
Lie group of higher rank is qu asiisometric to the r estrictio n of an y leftin v arian t riemannian metric of the lie group itself. Lie groups, algebraic groups and lattices alexander gorodnik abstract this is a brief introduction to the theories of lie groups, algebraic groups and their discrete subgroups, which is based on a lecture series given during the summer school held in the banach centre in poland in summer 2011. The problem of classifying the real reductive groups largely reduces to classifying the simple lie groups. We will discuss the classi cation of semisimple lie algebras, root systems, the weyl group, and dynkin diagrams. Pdf divergence in lattices in semisimple lie groups and. Quasiisometric rigidity of higher rank sarithmetic lattices kevin wortman we show that sarithmetic lattices in semisimple lie groups with no rank one factors are quasiisometrically rigid. Pdf dimension rigidity of lattices in semisimple lie groups.
Get a printable copy pdf file of the complete article 571k, or click on a page image below to browse page by page. In section 2, we will collect some general results on lattices in locally compact. Five lectures on lattices in semisimple lie groups 5 b1 a12a21 0. Also, the lie group r is reductive in this sense, since it can be viewed as the identity component of gl1,r. The purpose of this thesis is to calculate l2invariants of nonuniform lattices in semisimple lie groups using suitable compacti cations of locally symmetric spaces. Proving a conjecture posed in 5, we give very precise bounds for the congruence subgroup growth of arithmetic groups. In the introduction we gave some examples of qrank 1 lattices. In particular, every connected semisimple lie group meaning that its lie algebra is semisimple is reductive. Subgroup growth of lattices in semisimple lie groups. This is a brief introduction to the theories of lie groups, algebraic groups and. On the other hand, the universal cover of sl 2, r is not a real reductive group, even though its lie algebra is reductive, that is, the product of a semisimple lie algebra and an abelian lie algebra. Quasiisometric rigidity of higher rank sarithmetic lattices.
If g an algebraically simplyconnected lie group, g gsol ogss. Functional relations for zetafunctions of weight lattices of lie groups of type a3 yasushi komori, kohji matsumoto and hirofumi tsumura department of mathematics, rikkyo university, nishiikebukuro, toshimaku, tokyo 1718501. Filling in solvable groups and in lattices in semisimple groups. The discrete series of semisimple groups peter hochs september 5, 2019 abstract these notes contain some basic facts about discrete series representations of semisimple lie groups. In th e case of q rank 1 w e use the structure of asymptotic. The leitmotiv is how discrete structures approximate continuous ones, or vice versa. On the other hand, we conjecture that lattices in semisimple lie groups of higher rank always have linear divergence. Pdf subgroup growth of lattices in semisimple lie groups. It is known that their lling order is at most exponential 20. Nevertheless, there is an interesting dichotomy between rn and nn on the one hand, and hn and c1 on the other. Let cbe the field of complex numbers and fa vector space over c. By definition, all finite coverings of sl 2, r such as the metaplectic group are real reductive groups. The group theoretic way characterising of the cocompact lattices is that they contain. To do this we shall say that a sequence of lattices converges to a lattice.
A class of groups with similar properties with respect to lattices to real semisimple lie groups are semisimple algebraic groups over local fields of characteristic 0, for example the padic fields. Arithmetic aspects of selfsimilar groups 739 informally speaking, the restriction of g to t w is a copy of the action of h on t, which explains the name selfsimilar. Get pdf 125 kb abstract in this article we prove that the cocompactness of the arithmetic lattices in a connected semisimple real lie group is preserved if the lattices under. For the theory of orbit equivalence for actions of amenable groups see 4, and for actions of semisimple lie groups and their lattices, see 36, 37. Put g1 radk and g2 gg1, a connected semisimple lie group. Pdf twisted conjugacy classes in lattices in semisimple lie. The solvable groups are interesting as far as few things are known on their behaviour up to quasiisometry. Knapp, structure theory of semisimple lie groups, 1997. Further, the algebraic group is a product of factors and by modding out compact ones you are in the situation described above. Lattices in semisimple lie groups, and multipliers of group c algebras mohammed e.
The link between the above two categories of objects. One also often considers reductive groups, which include semisimple groups but are allowed to have a larger center, like gln,rorun. Browse other questions tagged group theory differentialgeometry lie groups lattices in lie groups or ask your own question. Generalizing further, any semisimple lie group g has transitive actions on. There is an arithmetic construction similar to the real case, and the dichotomy between higher rank and rank one. Root lattices form a special family of lattices admitting a large amount of symmetries. Notations and conventions let hbe a semisimple lie group without compact factors. This is a book about arithmetic subgroups of semisimple lie groups, which means that we will discuss the group sl n. If the group is simplyconnected, these zetafunctions coincide with ordinary zetafunctions of root systems of associa. Lie algebras are an essential tool in studying both algebraic groups and lie groups. Presentations for cusped arithmetic hyperbolic lattices.
L2invariants of nonuniform lattices in semisimple lie groups. Dec 18, 2003 on orispherical subgroups of a semisimple lie group maruyama, shigeya, kodai mathematical seminar reports, 1968. Edit back in time 1 revision see changes history cite print tex source. If g is a connected lie group, then a lattice in g is a discrete subgroup. We also study nonuniform lattices in semisimple groups. The case of real rank one, where the associated symmetric space is negatively curved, is of special. The length of conjugators in solvable groups and lattices of semisimple lie groups abstract. Divergence in lattices in semisimple lie groups and.
Zetafunctions of weight lattices of compact connected semisimple lie groups asushiy komori, kohji matsumoto, hirofumi tsumura abstract. The exact lling order is already known for almost all cases when the ambient semisimple group is of rrank one see the comments following remark 4. Ergodic theory of semisimple groups theorem margulis, nevo, stein. Heisenberg groups, the stabilizer is a nonuniform lattice in mg 2. Chapter i develops the basic theory of lie algebras, including the fundamental theorems of engel, lie, cartan, weyl, ado, and poincarebirkhoffwitt. For a large part, they summarise relevant material from knapps book 12. Discrete subgroups of real semisimple lie groups 557 let us introduce a topology into the space w, consisting of all lattices lying in some finitedimensional euclidean space r.
If you want to understand lattices in semisimple lie groups, then, indirectly, you see their relation to finite coxeter groups. If we look at an object such as a crystal with light whose wavelength is. Smooth classification of cartan actions of higher rank semisimple lie groups and their lattices pages 743773 from volume 150 1999, issue 3 by edward r. Five lectures on lattices in semisimple lie groups 3 p. In chapter iii, we give a geometric idea of the proof of the main theorem. Motivated by the above, reid asked whether or not lattices in rank one real lie groups are pro nitely rigid 14, q.
Bekka and alain valette 1 introduction and some history. Lie solvable groups endowed with a left invariant riemannian structure and nonuniform lattices in semisimple groups endowed with a left invariant word metric. Groups korea 94 proceedings of the international conference held at pusan national university, pusan, korea, august 1825, 1994 ed. Aspects of the geometric and algebraic structure of lattices will be discussed, and the superrigidity and arithmeticity theorems of margulis will be described. Nevertheless, there is an interesting dichotomy between rn and nn on. The length of conjugators in solvable groups and lattices of. Stationary characters on lattices of semisimple lie groups. However, it also has important applications in geometry. Lattices and lie algebras laura geatti the purpose of these lectures is to explain the connection between complex semisimple lie algebras and root lattices. Invariant eigendistributions on semisimple lie groups harishchandra, bulletin of the american mathematical society, 1963. We refer to, or for a definition of qrank 1 lattices in semisimple groups. More precisely let h be a connected semisimple lie group and.
Viewingx g l as a twosided ideal in the measure algebr mga, and viewing elements of l1h as. Ghys gh recently proved that irreducible lattices in most semisimple lie groups of higher real rank do not have any interesting differentiable actions on the circle t. G connected semisimple lie group with no compact factors, g t kbiinvariant riemannian balls ing, s tfx 1 mg t r g t fg. Let i\ be a lattice in a simple lie group g with r7wg2, and let fg be the fundamental group of a compact manifold of. Subgroup growth of lattices in semisimple lie groups lubotzky, alexander and nikolov, nikolay, acta mathematica, 2004 the adjusted viterbi training for hidden markov models lember, juri and koloydenko, alexey, bernoulli, 2008. Isoperimetric inequalities for lattices in semisimple lie groups of rank 2. This is a book about arithmetic subgroups of semisimple lie groups, which means that we will discuss the group sln. Similarly, their geometric counterpart, the closed locally symmetric spaces of noncompact type, form the main examples of nonpositively curved manifolds. Lattices are not solvable in noncompact semisimple lie groups. Lattices in lie groups school of mathematics, tifr. We prove it in the case when the qrank is 1 and when the lattice is slnos where. Next we will discuss some generalities about lie algebras. Marguliss normal subgroup theorem a short introduction. Strong rigidity for ergodic actions of semisimple lie groups.
Divergence in lattices in semisimple lie groups and graphs. On the other hand, every element of infinite order in a finitely generated abelian group is undistorted. For semisimple lie groups, using the polar coordinates on semisimple group, it can be easily established that any bikinvariant riemannian metric k a maximal compact subgroup have strict t q exp ct volume growth, with q and c 0 depending on the metric. Filling in solvable groups and in lattices in semisimple. Lie groups, algebraic groups and lattices institut fur mathematik. Examples will be provided by simple arithmetic constructions.
As mentioned above, by margulis arithmeticit y theorem 16 every lattice. A lattice in a semisimple lie group of higher rank has to contain distorted elements of infinite order a theorem of lubotzky, mozes and raghunathan. Divergence in lattices in semisimple lie groups and graphs of groups article pdf available in transactions of the american mathematical society 3625 january 2008 with 37 reads. There is an arithmetic construction similar to the real case, and the dichotomy between higher rank and rank one also holds in this case, in a more marked form. There are several books dealing with lattices in semisimple lie groups, the gentlest one, i think, is introduction to arithmetic groups by dave witte morris.