Texas Geometry and Topology Conference This is a report on the presentations at the 37th meeting of the Texas Geometry and Topology Conference at the Texas Christian University, March 23-25, 2007. This conference was partially supported by National Science Foundation Grant DMS-0605082 and Texas Christian University. Speakers reported on recent re- search. For this report, speakers have provided synopses of their talks together with broader discussions of the significance and context of their results. Meeting 37. Texas Christian University, March 23-25, 2007 Tara Brendle, Louisiana State University, Wicket groups and ring groups 3 Let Wn be the space of all configurations of n disjoint semicircles in upper halfspace R+ that meet the plane 3 2 @R+ = R orthogonally in their endpoints. We call these semicircles wickets, as in the game of croquet. Our main purpose is to study the wicket group Wn = π1(Wn) and relate it to other better-known groups. 2 By taking the union of a configuration of wickets in Wn with its reflection across the plane R we obtain a configuration of n pairwise unlinked circles, so Wn embeds in the space Rn of all configurations of n 3 disjoint pairwise unlinked circles, or rings,inR . We will also be interested in the ring group Rn = π1(Rn). The inclusion Wn ,→Rninduces a homomorphism Wn → Rn whichweshowissurjective.ThegroupRn (as well as finite index subgroups of Rn obtained by orienting and perhaps also numbering the rings) has been studied before in a number of different guises under such different names as the ‘motion group’ of the unlink [D], [Go], [BL], [R], the ‘symmetric automorphism group’ of a free group [C], [M], [BMMM], the ‘group of loops’ [JMM], the ‘braid-permutation group’ [FRR], and the ’loop braid group’ [BWC]. Another term that is often used is the ‘string group’. The connection between Wn and braid groups arises from associating to each configuration in Wn the endpoints of its n semicircles, a configuration of 2n points in R2. Passing to loops of such configurations, we obtain a homomorphism from Wn to the braid group B2n. This homomorphism turns out to be injective, so Wn embeds as a subgorup of B2n. There is a simpler space of wicket configurations, the subspace Un of Wn consisting of configurations of wickets lying in planes parallel to a fixed plane, say the xz-plane. We think of these as ‘untwisted’ wickets, and we set Un = π1(Un). We show that the inclusion of Un into Wn induces an injection Un ,→ Wn and that Un is a K(Un; 1). We are then able to use a general position argument to produce an explicit finite presentation for Un. A little further effort yields presentations for Wn and for Rn, each obtained by adding certain ‘obvi- ous’ generators and relations to that of Un. To this end, one technical tool we introduce is the ‘separated wicket’ subspace SWn of Wn consisting of configurations of wickets which can be encased in hemispheri- cal ‘shells’ in a particular way. In a similar fashion we define a subspace SRn of Rn consisting of separated configurations of rings. Theorem. The inclusions SWn ,→Wn and SRn ,→Rn are homotopy equivalences. There are also ‘floppy’ versions of wicket space and ring space, namely arc space An and loop space Ln, respectively. More precisely, An is the space of all configurations of n disjoint smoothly embedded 3 3 unknotted arcs in R+ which intersect @R+ transversely in endpoints; Ln is then defined in the obvious way. We show the following. Theorem. The inclusions Wn ,→Anand Rn ,→Lnare homotopy equivalences. Generators for An = π1(An) were found by Hilden [Hi]. Our results give a new derivation of these generators, along with a full set of relations for them. We note that in Hilden’s viewpoint, An embeds Spring 2007 TGTC Report Page 1 of 10 in a natural way in the so-called Heegaard subgroup of the mapping class group of a surface. Also, the group π1(Ln) was computed in the 1960s by Dahm [D] by showing that the natural map from π1(Ln) 3 to the automorphism group of the free group π1(R − L) for L ∈Lnis injective, with image the so- called symmetric automorphisms, the subgroup generated by automorphisms sending each basis element to a conjugate of itself, along with the automorphisms that permute basis elements or send them to their inverses. The braid group Bn is defined as the fundamental group of the space Cn of configurations of n points 2 in R , and it is an elementary fact that the higher homotopy groups of Cn are trivial, so it is a K(Bn; 1). Analogously, we show the following. Theorem. The map Wn → B2n induced by the endpoint map Wn →C2nis injective, and Wn is a K(Wn; 1). The proof of this theorem is considerably more involved than for Cn. Note also that Rn is not a K(Rn; 1), and indeed there is no finite-dimensional K(Rn; 1),sinceRn has torsion, as it contains a copy of the symmetric group Sn. References [BWC] J. C. Baez, D. K. Wise, A. S. Crans, Exotic statistics for strings in 4D BF theory, preprint (2006), arXiv:gr- qc/0603085. [BMMM] N. Brady, J. McCammond, J. Meier, and A. Miller, the Pure symmetric automorphisms of a free group form a duality group, J. Algebra, 246 (2001), no. 2, 881–896. [BL] A. Brownstein and R. Lee, Cohomology of the group of motions of n strings in 3-space. In Mapping Class Groups and Moduli Spaces of Riemann Surfaces,G¨ottingen (1991), Seattle (1991), Contemp. Math., 150 (1993), 51–61. [C] D. J. Collins, Cohomological dimension and symmetric automorphisms of a free group, Comment. Math. Helv., 64 (1989), 44–61. [D] D. M. Dahm, A generalization of braid theory, Ph. D. Thesis, Princeton University (1962). [FRR] R. Fenn, R. Rim´anyi, and C. Rourke, The braid permutation group, Topology, 36 (1997), 123–135. [Go] D. L. Goldsmith, The theory of motion groups. Michigan Math. J., 28 (1981), 3–17. [Hi] H. Hilden, Generators for two groups related to the braid group, Pac. J. Math., 59 (1975), 475–486. [JMM] C. Jensen, J. McCammond, and J. Meier, The integral cohomology of the group of loops, Geom. & Top., 10 (2006), 759–784. [M] J. McCool, On basis-conjugating automorphisms of the free groups, Can. J. Math., 38 (1986), 1525–1529. [R] R. L. Rubinsztein, On the group of motions of oriented, unlinked and unknotted circles in R3, I, preprint, Uppsala University (2002), (http://www.math.uu.se/research/pub/Rubinsztein1.pdf). [V] V. V. Vershinin, On homological properties of singular braids, Trans. A.M.S., 350 (1998), 2431–2455. Spring 2007 TGTC Report Page 2 of 10 Richard Canary, University of Michigan, Introductory bumponomics: the deformation theory of Kleinian groups We study the topology of the space AH(M) of all (marked) hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M (with boundary.) The topology of the interior of AH(M) is quite simple and has been well-understood since the 1970’s. The components are enumerated by (marked) homeomorphism types of compact hyperbolizable 3-manifolds homotopy equivalent to M and each component is a manifold which is naturally identified with a quotient of a Teichm¨uller space. In the last decade it has become clear that the global topology of AH(M) is quite complicated. An- derson and Canary discovered that components of the interior can “bump” (i.e. have intersecting closures). Anderson, Canary and McCullough completely characterized which components of the interior of AH(M) bump when M has incompressible boundary. McMullen gave the first example where a single component of the interior of AH(M) self-bumps, i.e there is a point in the boundary of the component such that the inter- section of any small enough neighborhood of the point with the component is disconnected. Bromberg and Holt showed that self-bumping is a very widespread phenomena, in particular any component of the interior of AH(M) self-bumps whenever M contains a primitive essential annulus. Most disturbingly, Bromberg proved that the space of punctured torus groups is not even locally connected. In this talk, we discuss recent joint work with Brock, Bromberg and Minsky, which shows, in many cases, that the topology is well-behaved at “most” points in the boundary of AH(M). We say that a point in the boundary @AH(M) of AH(M) is untouchable if there is no bumping or self-bumping of components at the point. In particular, AH(M) is locally connected at all untouchable points. Theorem 1. If N ∈ @AH(M) has no cusps (i.e. every homotopically non-trivial closed curve in N is homotopic to a closed geodesic), then N is untouchable. If @M contains no tori, then such points are generic in @AH(M). Notice that we do not assume in this result that @M has incompressible boundary. We say that N ∈ @AH(M) is quasiconformally rigid if every component of its conformal boundary is a thrice-punctured sphere. Theorem 2. If N ∈ @AH(M) is quasiconformally rigid, then there is no bumping at N. We further say that N is a maximal cusp if N is quasiconformally rigid and geometrically finite. Theorem 3. If M is acylindrical and N ∈ @AH(M) is a maximal cusp, then there is no self-bumping at N. Steve Ferry, Rutgers University, Higson’s compactification and the Novikov conjecture In 1990, Nigel Higson introduced a very general compactification, reminiscent of the Stone-Cech compact- ification, for locally compact metric spaces.