We determine the mod $2$ cohomology over the Steenrod algebra of the classifying spaces of the free loop groups $LG$ for compact groups $G=Spin(7)$, $Spin(8)$, $Spin(9)$, and $F_4$. Then, we show that they are isomorphic as algebras over the Steenrod algebra to the mod $2$ cohomology of the corresponding Chevalley groups of type $G(q)$, where $q$ is an odd prime power. In a similar manner, we compute the cohomology of the free loop space over $BDI(4)$ and show that it is isomorphic to that of $BSol(q)$ as algebras over the Steenrod algebra.
We prove that various subgroups of the mapping class group $Mod(Sigma)$ of a surface $Sigma$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstadt), the point-pushing and surface braid subgroups, and the Lagrangian subgroup. Our techniques include a method to compute lower bounds on distortion via representation theory and an extension of Johnson theory to arbitrary subgroups of $H_1(Sigma;mathbb{Z})$.
We give a unified solution to the conjugacy problem for Thompsons groups F, T, and V. The solution uses strand diagrams, which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompsons groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompsons groups, and we use this correspondence to investigate the dynamics of elements of F. Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results.
Let ${cal M}_{g,n}$ and ${cal H}_{g,n}$, for $2g-2+n>0$, be, respectively, the moduli stack of $n$-pointed, genus $g$ smooth curves and its closed substack consisting of hyperelliptic curves. Their topological fundamental groups can be identified, respectively, with $Gamma_{g,n}$ and $H_{g,n}$, the so called Teichm{u}ller modular group and hyperelliptic modular group. A choice of base point on ${cal H}_{g,n}$ defines a monomorphism $H_{g,n}hookrightarrowGamma_{g,n}$. Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichmuller group $Gamma_{g,n}$ is the group of isotopy classes of diffeomorphisms of the surface $S_{g,n}$ which preserve the orientation and a given order of the punctures. As a subgroup of $Gamma_{g,n}$, the hyperelliptic modular group then admits a natural faithful representation $H_{g,n}hookrightarrowoperatorname{Out}(pi_1(S_{g,n}))$. The congruence subgroup problem for $H_{g,n}$ asks whether, for any given finite index subgroup $H^lambda$ of $H_{g,n}$, there exists a finite index characteristic subgroup $K$ of $pi_1(S_{g,n})$ such that the kernel of the induced representation $H_{g,n}tooperatorname{Out}(pi_1(S_{g,n})/K)$ is contained in $H^lambda$. The main result of the paper is an affirmative answer to this question for $ngeq 1$.
We deal with some pcf investigations mostly motivated by abelian group theory problems and deal their applications to test problems (we expect reasonably wide applications). We prove almost always the existence of aleph_omega-free abelian groups with trivial dual, i.e. no non-trivial homomorphisms to the integers. This relies on investigation of pcf; more specifically, for this we prove that almost always there are F subseteq lambda^kappa which are quite free and has black boxes. The almost always means that there are strong restrictions on cardinal arithmetic if the universe fails this, the restrictions are everywhere. Also we replace Abelian groups by R-modules, so in some sense our advantage over earlier results becomes clearer.
We study the Weyl groups of hyperbolic Kac-Moody algebras of `over-extended type and ranks 3, 4, 6 and 10, which are intimately linked with the four normed division algebras K=R,C,H,O, respectively. A crucial role is played by integral lattices of the division algebras and associated discrete matrix groups. Our findings can be summarized by saying that the even subgroups, W^+, of the Kac-Moody Weyl groups, W, are isomorphic to generalized modular groups over K for the simply laced algebras, and to certain finite extensions thereof for the non-simply laced algebras. This hints at an extended theory of modular forms and functions.
Divergence functions of a metric space estimate the length of a path connecting two points $A$, $B$ at distance $le n$ avoiding a large enough ball around a third point $C$. We characterize groups with non-linear divergence functions as groups having cut-points in their asymptotic cones. By Olshanskii-Osin-Sapir, that property is weaker than the property of having Morse (rank 1) quasi-geodesics. Using our characterization of Morse quasi-geodesics, we give a new proof of the theorem of Farb-Kaimanovich-Masur that states that mapping class groups cannot contain copies of irreducible lattices in semi-simple Lie groups of higher ranks. It also gives a generalization of the result of Birman-Lubotzky-McCarthy about solvable subgroups of mapping class groups not covered by the Tits alternative of Ivanov and McCarthy. We show that any group acting acylindrically on a simplicial tree or a locally compact hyperbolic graph always has many periodic Morse quasi-geodesics (i.e. Morse elements), so its divergence functions are never linear. We also show that the same result holds in many cases when the hyperbolic graph satisfies Bowditchs properties that are weaker than local compactness. This gives a new proof of Behrstocks result that every pseudo-Anosov element in a mapping class group is Morse. On the other hand, we conjecture that lattices in semi-simple Lie groups of higher rank always have linear divergence. We prove it in the case when the $mathbb{Q}$-rank is 1 and when the lattice is $SL_n(mathcal{O}_S)$ where $nge 3$, $S$ is a finite set of valuations of a number field $K$ including all infinite valuations, and $mathcal{O}_S$ is the corresponding ring of $S$-integers.
We calculate the abelianizations of the level $L$ subgroup of the genus $g$ mapping class group and the level $L$ congruence subgroup of the $2g times 2g$ symplectic group for $L$ odd and $g geq 3$.
These are the lecture notes of a 2-hour mini-course on Lie groups over local fields presented at the Workshop on Totally Disconnected Groups, Graphs and Geometry at the Heinrich-Fabri-Institut Blaubeuren in May 2007. The goal of the notes is to provide an introduction to p-adic Lie groups and Lie groups over fields of formal Laurent series, with an emphasis on relations to the structure theory of totally disconnected, locally compact groups. In particular, they contain a discussion of the scale, tidy subgroups and contraction groups for automorphisms of Lie groups over local fields. Special attention is paid to the case of Lie groups over local fields of positive characteristic.
We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G, then for almost every x in G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapioni-Serra Cassano in step 2 Carnot groups: they have shown that, for almost every x, E has a unique tangent at x, and this tangent is a vertical halfspace.
