Hyperbolic rigidity of higher rank lattices

We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of right-angled groups. A group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute. Most nonuniform lattices in higher rank simple Lie groups are right angled. We provide the first examples of uniform (cocompact) right-angled arithmetic groups in SL(n,R), n≥3, and SO(p,q) for some values of p,q. This is a class of lattices for which the congruence subgroup property is not known in general. By using rigidity theory and the notion of invariant random subgroups it follows that both the rank gradient and the homology torsion growth vanish for an arbitrary sequence of subgroups in any right-angled lattice in a higher rank simple Lie group.

Let Mo,, denote the moduli space of Riemann spheres with n ordered marked points. In this article we define the group Out# of quasispecial symmetric outer automorphisms of the algebraic fundamental group 1r1(Mo,n) for all n > 4 to be the group of outer automorphisms respecting the conjugacy classes of the inertia subgroups of Irl(M4,n) and commuting with the group of outer automorphisms of rl (MO,n) obtained by permuting the marked points. Our main result states that Outd is isomorphic to the Grothendieck-Teichmiiller group GT for all n > 5.

The canonical modular homomorphism provides an embedding of R \mathbb {R} into the outer automorphism group O u t ( M ) Out(M) of any type I I I 1 \mathrm {III}_{1} factor M M . We provide an explicit construction of a full factor of type I I I 1 \mathrm {III}_{1} with separable predual such that the outer automorphism group is minimal, i.e. this embedding is an isomorphism and a homeomorphism. We obtain such a I I I 1 \mathrm {III}_{1} factor by using an amalgamated free product construction.

If K G Q is a group extension, then any automorphism of G which sends K into itself, induces automorphisms respectively on K and on Q. This subgroup of automorphisms of G is denoted by Aut(G; K) and is called the automorphism group of the extension K G Q. After establishing an interesting group action of Aut(K) Aut(Q) on the set H 2 (Q; K) of all 2cohomology classes of Q with coecien ts in K, a full description of Aut(G; K) and Out(G; K) = Aut(G; K)= Inn(G) is obtained in terms of various commutative diagrams. This picture is as general as possible, hence covering and further complementing similar ideas developed earlier by C. Wells ([5]), P. Conner & F. Raymond ([1]), D.J.S. Robinson ([3], [4]) and the author ([2]). 1 Notations and preliminaries If G is a group and x 2 G, then (x) is the inner automorphism determined by x (sending y 2 G to xyx 1 ), (G) is known as the inner automorphism group Inn(G) and Out(G) = Aut(G)= Inn(G) is called the outer automorphism group of G. Write p : Aut(G) Out(G) for the natural projection. For a subset X in G, CGX denotes the centralizer and NGX is the normalizer of X in G. Let Z(G) be the center of G. In the sequel of this paper, aspects of group cohomology (with non-abelian coecien ts) will be intensively used. Therefore, we review some basic facts of this theory and meanwhile x additional notations and terminology.

In this paper we show that the groups of automorphisms and outer automorphisms of a finitely generated free group have isoperimetric functions which are bounded above by exponential functions. This exponential bound is best possible if the free group has rank three, but the best bound remains unknown in higher rank. Our techniques show more generally that n-dimensional isoperimetric functions for these groups are at most exponential for all n. A variation of the technique gives an asynchronous bounded combing of the mapping class group of a bounded surface.

Twisted first homology groups of the automorphism group of a free group

Verification of Dade's Conjecture for Janko GroupJ3

group of odd order has only inner automorphisms. The group of automorphisms of the holomorph of an arbitrary group was studied by Gol'fand [3] who found some cases in which the outer automorphism group has order one or two. In the present paper I determine explicitly the outer automorphism group 0 of the holomorph H of an arbitrary finite abelian group G. If G is the direct product of a group of odd order, a group of order two, and a cyclic group of order 2n where n> 2, then 0 is the direct product of a finite number of groups of order two and a non-abelian group 0* of order six or eight. If n =2 then 0 * is isomorphic to the symmetric group of order six, and if n >3 then 0* is the octic group. In all other cases 0 is either trivial or the direct product of a finite number of groups of order two. Let A be the group of all automorphisms of the finite abelian group G, let GB be the group of all automorphisms of H that map G onto itself, and let g be the group of all inner automorphisms of H. Then (B3/J can be identified with the first cohomology group H'(A, G). Thus H'(A, G) can be regarded as a subgroup of 0. Now G is an ilnvariant subgroup of H, and it is known [6] that H has at most four invariant subgroups isomorphic to G. It follows that H'(A, G) has index at most four in 0.

The geometric dimension for proper actions \underline{\mathrm{gd}}(G) of a group G is the minimal dimension of a classifying space for proper actions \underline{E}G . We construct for every integer r\geq 1 , an example of a virtually torsion-free Gromov-hyperbolic group G such that for every group \Gamma which contains G as a finite index normal subgroup, the virtual cohomological dimension vcd (\Gamma) of \Gamma equals \underline{\mathrm{gd}}(\Gamma) but such that the outer automorphism group Out (G) is virtually torsion-free, admits a cocompact model for \underline{E} Out (G) but nonetheless has vcd(Out (G))\le \underline{\mathrm{gd}} (Out (G))-r .

Let p , l be distinct prime numbers. A tripod-degree over p at l is defined to be an l -adic unit obtained by forming the image, by the l -adic cyclotomic character, of some continuous automorphism of the geometrically pro- l fundamental group of a split tripod over a finite field of characteristic p . The notion of a tripod-degree plays an important role in the study of the geometrically pro- l anabelian geometry of hyperbolic curves over finite fields, e.g., in the theory of cuspidalizations of the geometrically pro- l fundamental groups of hyperbolic curves over finite fields. In the present paper, we study the tripod-degrees. In particular, we prove that, under a certain condition, the group of tripod-degrees over p at l coincides with the closed subgroup of the group of l -adic units topologically generated by p . As an application of this result, we also conclude that, under a certain condition, the natural homomorphism from the group of automorphisms of the split tripod to the group of outer continuous automorphisms of the geometrically pro- l fundamental group of the split tripod that lie over the identity automorphism of the absolute Galois group of the basefield is surjective.

We prove that an outer action of a locally compact group G on a full factor M is automatically strictly outer, meaning that the relative commutant of M in the crossed product is trivial. If moreover the image of G in the outer automorphism group Out M is closed, we prove that the crossed product remains full. We obtain this result by proving that the inclusion of M in the crossed product automatically has a spectral gap property. Such results had only been proven for actions of discrete groups and for actions of compact groups, by using quite different methods in both cases. Even for the canonical free Bogoljubov actions on free group factors or free Araki–Woods factors, these results are new.

We prove that certain 1-relator groups have Property E. Using this fact, we characterize all conjugacy separable 1-relator groups of the form 〈a, b; (a -α b β a α b γ) t 〉, t ⩾ 1, having residually finite outer automorphism groups.

A rigid automorphism of a linking system is an automorphism that restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $2$ is elementary abelian and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group G restricts to the identity on the centric linking system for G, then it is of $p'$ -order modulo the group of inner automorphisms, provided G has no nontrivial normal $p'$ -subgroups. We present two applications of this last result, one to tame fusion systems.

We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various sub- groups of the mapping class group—the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration—and additionally to the outer automorphism group of a free group and to certain linear groups. There are many analogies and interconnections between the theories of right- angled Artin groups on one hand and mapping class groups on the other hand. For instance, by the work of Crisp and Wiest (13), the work of Koberda (27), and the work of the first two authors with Mangahas (12), there is an abundance of injective homomorphisms from right-angled Artin groups to mapping class groups. Also, the last two authors proved (30) that any two elements of the pure braid group either generate a free group or a free abelian group—a property shared by all right-angled Artin groups (4, Theorem 1.2). We are thus led to ask to what extent mapping class groups are the same as right-angled Artin groups. It is straightforward to see that most mapping class groups are not isomorphic to right-angled Artin groups, for instance because right-angled Artin groups are torsion free. On the other hand, mapping class groups have finite-index sub- groups that are torsion free, and so this leaves open the possibility that mapping class groups are abstractly commensurable to right-angled Artin groups, that is, that they have isomorphic finite-index subgroups. We prove that, aside from a small number of exceptions, this is not the case. We also extend this result to several classes of groups related to mapping class groups. We start by recalling some definitions. To a finite graph , we can associate a right-angled Artin grou p: this is the group with one generator for each vertex of , and one defin ing relator for each edge, namely, the commutator of the two generators corresponding to the endpoints. Let Sg,n denote a closed, connected, orientable surface of genus g with n marked points. The mapping class group Mod(Sg,n) is the group of homotopy classes of orientation-preserving homeomorphisms of Sg,n preserving the set of marked points. As discussed in Koberda's paper (27, Theorem 1.5), no finite-index subgroup of Mod(Sg,n) injects into a right-angled Artin group if g ≥ 2 and (g,n) 6 (2,0); see

Given L > 0 elements in a word hyperbolic group G, there exists a number M = M(G, L) > 0 such that at least one of the assertions is true: (i) these elements generate a free and quasiconvex subgroup of G; (ii) they are Nielsen equivalent to a system of L elements containing an element of length at most M up to conjugation in G. The constant M is given explicitly. The result is generalized to groups acting by isometries on Gromov hyperbolic spaces. For proof we use a graph method to represent finitely generated subgroups of a group.