최근 연구 내용 소개/Recent Research Topics

백형렬/Hyungryul Baik

Unsmoothability of mapping class group actions on compact one-manifolds

1차원 다양체 대한 mapping class group의 작용이 smooth할 수 있는가에 관한 연구

곡면이라는 것은 2차원 다양체이다. 우리는 그 중에서도 사람들이 finite-type이라고 일컫는 2차원 다양체를 생각하려고 한다. 즉, 콤팩트하고 유한 개의 경계면을 가지는 2차원 다양체에 유한 개의 구멍을 뚫어서 만들어지는 경우이다.

어떤 곡면이 주어졌일 때, 그 곡면에서 자기 자신으로 가는 homeomorphism들을 모아놓아 놓으면 두 개의 homeomorphism의 합성 함수가 다시 homeomorphism이 되므로 group을 형성하는 것을 알 수 있다. 여기서 두 개의 homeomorphism이 isotopic하면 (homotopy는 continuous map들을 통한 변화를 의미한다면, isotopy는 매 순간 homeomorphism이도록 하면서 변화시키는 것이다) 같은 homeomorphism으로 보겠다고 하여도, 계속해서 group이 되는 것을 쉽게 보일 수 있는데, 이렇게 곡면 S의 homeomorphism들의 isotopy class들을 모아 놓은 group을 mapping class group of S라고 부르고, Mod(S)라고 쓴다. 한 가지 주의할 것은, 이 곡면이 닫힌 곡면이 아닐 때는 isotopy가 정확히 어떻게 정의되는가를 정해야하는데, 보통은 경계면의 모든 점들을 고정하는 isotopy들을 생각한다.

Nielsen은 이미 곡면 S가 경계면은 없고 구멍만 한 개 뚫려있는 경우에 mapping class group, Mod(S)가 원에 continuous action이 있다는 것을 보였다. 여기서 action은 faithful action만을 의미하는데 그것은 group의 두 개의 서로 다른 원소가 임의로 주어지면 그들의 작용이 완전히 일치하지는 않는다는 것을 의미한다. 예를들어 group은 임의의 공간에 작용을 하는데, 그냥 모든 원소가 모든 점을 고정하는, 즉 `아무일도 하지 않는' 작용을 생각할 수 있기 때문이다. 이러한 작용은 전혀 흥미로운 결과를 주지 못한다.

그렇다면 Mod(S)의 action이 continuous action이 아니라 smooth action일 수 있는 지에 대해 생각해볼 수 있다. 이 문제는 Nielsen 이후 오랜 시간 풀리지 않고 있었는데, Farb와 Franks[3]가 2010년에 경계면이 없고 구멍이 없거나 한 개만 있는 거의 모든 곡면 S에 대해서는 Mod(S)가 원에 smooth action이 있을 수 없다는 것을 보였다 (Ghys도 이 결과를 독립적으로 얻었다고 한다).

서울대학교의 김상현 교수님과 미국 버지니아 대학교의 Thomas Koberda교수과의 공동연구[2]를 통해 이 질문에 대한 답을 찾았는데, 그것은 거의 모든 곡면 S에 대해 Mod(S)가 원에 대한 smooth 작용을 할 수 없다는 것이다. 뿐만 아니라 Mod(S)의 유한 지수를 가지는 임의의 부분군을 생각하여도 같은 결과를 얻는다. 이것은 직교아티군이 원에 언제 smooth하게 작용할 없는지를 밝히는 방법으로 증명하였다. 참고로 직교아티군은 실수측에 대해서는 늘 smooth하게 작용할 수 있다는 사실[1]이 있는데 원에서는 결과는 이와 아주 대조적임을 알 수 있다. 더 정확한 결과를 알기 위해서는 아래의 영문 설명을 읽어보기를 바란다.

s For an orientable surface S = Sg,n,b of genus g with n punctures and b boundary components, it is a question with long history if the mapping class group Mod(S) (the group of isotopy classes of self-homeomorphisms of S) admits a smooth action on S. It is a classical fact due to J. Nielsen that Mod(Sg,1,0) admits a faithful continuous action on S1. B. Farb and J. Franks [3] showed that Mod(Sg,n,0) with g 3 and n ≤ 1 admits no nontrivial C2-action on a compact 1-dimensional manifold (this is independently due to E. Ghys in the case of S1). With S. Kim and T. Koberda, we gave a complete answer to this question in [2]. We showed that there exists a finite index subgroup of Mod(S) which admits a faithful C{1+bv}-action on a compact 1-manifold if and only if c(S) ≤ 1 where C{1+bv}-homeomorphism is a C1 homeomorphism whose first derivative has bounded variation, and c(S) = 3g-3+n+b. This result is proved by showing that no finite index subgroup of the right-angled Artin group (RAAG) of the path graph with 4 vertices admits a faithful C{1+bv}-action on a compact 1-manifold. This is dramatically different from the case of non-compact 1-manifolds, since every RAAG admits a C∞-action on the real line [1].

[1] H. Baik, S. Kim and T. Koberda, Right-angled Artin subgroups of the C∞ diffeomorphism group of the real line, Israel J. Math. (2016), Volume 213, Issue 1, pp. 175–182.
[2] H. Baik, S. Kim and T. Koberda, Unsmoothable group actions on compact one-manifolds (with S. Kim and T. Koberda), to appear in J. Eur. Math. Soc.
[3] B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds, I: actions of nonlinear groups, arXiv:math/0107085.

Characterization of surface groups (Fuchsian groups) and 3-manifold groups in terms of their actions on S1

곡면과 3차원 다양체 기본 군들을 원에 대한 작용으로 특정지을 수 있는가에 관한 연구

One can identify Isom+(H2) with a subgroup of Homeo+(S1) by realizing S1 as the ideal boundary of H2. A Fuchsian group is a torsion- free discrete subgroup of Homeo+(S1) which is conjugate to a subgroup of Isom+(H2). A lamination is a set of unordered pairs of points of S1 which can be obtained as the set of endpoints of leaves of some geodesic lamination on H2. A basic hyperbolic geometry argument can show that Fuchsian groups have many invariant laminations on S1. In [2], we showed that having many invariant laminations is in fact a characterizing property of Fuchsian groups. More precisely, it is proved in [2] that a torsion-free discrete subgroup of Homeo+(S1) is Fuchsian if and only if it preserves three different very full invariant laminations. We also proposed a conjectural characterization of fibered 3-manifold groups. The conjecture is that a torsion-free discrete subgroup of Homeo+(S1) is a fibered 3-manifold group if and only if it preserves two very full loose invariant laminations with no additional invariant laminations. Many partial evidences have been obtained in [2] and [1].

[1] J. Alonso, H. Baik and E. Samperton, On Laminar groups, Tits alternatives, and Convergence group actions on S2, arXiv:1411.3532.
[2] H. Baik, Fuchsian Groups, Circularly Ordered Groups, and Dense Invariant Laminations on the Circle, Geom. Topol. (2015), 19-4, pp. 2081–2115.

Smallest positive eigenvalue of Laplace-Beltrami operators on fibered 3-manifolds

원 위의 파이퍼 번들로 나타나는 3차원 다양체 상에서 라플라스-벨트라미 연산자의 고유값에 관한 연구

We studied the smallest positive eigenvalue λ1(M) of the Laplace-Beltrami operator on a closed hyperbolic 3-manifold M which fibers over the circle. In general, M fibers in many different ways, and let genus of M be the minimal genus of a fiber. By combining the results of [2], [3] and [5], there exists a constant C only depending on the genus of M such that 1/(C vol(M)2) ≤ λ1(M) ≤ C/vol(M). In [1], we show that λ1(M) ≤ C log vol(M)/(vol(M)(2^(2g−2)/(2^(2g−2)−1)) where g is the genus of M and C is a constant only depending on g. This bound is essentially sharp in the sense that there exists a sequence (Mi) of fibered 3-manifolds of genus gi such that λ1(Mi) ≤ C log vol(Mi)/(vol(Mi)(2^(2g_i−2)/(2^(2g_i−2)−1)) as constructed in [1]. A similar result was independently announced by A. Lenzhen and J. Souto. On the other hand, it is also shown in [1] that random fibered 3-manifolds of genus g satisfy λ1(M) ≤ C/vol(M)2 for some constant C depending on C (disproving a conjecture of I. Rivin [4]), i.e., if Mn is the fibered 3-manifold whose monodromy is the n-th step of the random walk on Mod(S), then the probability that λ1(Mn) ≤ C/vol(Mn)2 approaches to 1 as n increases to infinity. In [1] the same result is obtained for typical fibered 3-manifolds where one uses, instead of the random walk, the proportion in the ball with respect to the word norm of Mod(S) and increases the size of the ball.

[1] H. Baik, I. Gekhtman and U. Hamenstädt, The smallest positive eigenvalue of fibered hyperbolic 3-manifolds, arXiv:1608.07609.
[2] P. Buser, A note on the isoperimetric constant. Ann. Scient. Ec. Norm. Sup. 15 (1982), 213–230.
[3] M. Lackenby, Heegaard splittings, the virtually Haken conjecture and Property (τ). Invent. Math. 164 (2006), 317–369.
[4] I. Rivin, Statistics of Random 3-Manifolds occasionally fibering over the circle. arXiv:1401.5736.
[5] R. Schoen, A lower bound for the first eigenvalue of a negatively curved manifold J. Differential Geom. 17 (2) (1982), pp.233-238.

Circular orders on groups

군의 원형 순서에 관한 연구

A circular order (a.k.a. cyclic order) on a set is an order relation on the set of triples of points which allows to tell if a given triple of elements are clockwise oriented or anti-clockwise oriented. On a group, we only consider a circular order which is invariant under the left-multiplication action of itself. For a countable group, the existence of such a left-invariant circular order is equivalent to the existence of a faithful action on S1 by orientation preserving homeomorphisms. In [1], we developed the theory of left-invariant circular orders of groups from a dynamical perspective following Navas’ work on linear orders of groups [2]. Using this theory, we classified all possible circular orders of finitely generated abelian groups, proved that the sets of non-linearly and non-circularly orderable finitely presented groups are recursively enumerable, provided a partial criterion for a given circular order to be isolated, and found a necessary and sufficient condition for a circular order so that the corresponding action on S1 is free.

[1] H. Baik and E. Samperton, Space of Circular Orders of Groups, arXiv:1508.02661, to appear in Groups Geom. Dyn.
[2] A. Navas. On the dynamics of (left) orderable groups. In Annales de l’institut Fourier, volume 60, pages 1685–1740. Association des annales de l’institut Fourier, 2010.

Torsion growth in the tower of covers of a random 3-manifold

임의의 3차원 다양체 덮개들로 만들어진 타워 상에서 호몰로지 군의 유한원소들의 갯수가 어떻게 증가하는가에 관한 연구

Given a manifold M and a tower of finite covers (Mi) of M, one can ask about the growth of topological invariants for the manifolds in the sequence. One interesting such invariant is the torsion in the first homology groups. As it became apparent in recent years, the existence of towers of covers with exponential torsion homology growth should be abundant for 3-manifolds (i.e., the rank of the torsion part of the first homology grows exponentially with respect to the degree of the covers). In the case of a tower of congruence covers of a closed arithmetic hyperbolic 3-manifold, there is a more precise conjecture [2] which relates the homology torsion growth with the l2-torsion of H3. In [1] we studied the 3-manifold obtained by gluing two handlebodies of genus g > 2 with a random element in the Torelli subgroup of Mod(Sg) (a random 3-manifold of genus g with maximal homology rank). In this case, the probability of such a manifold having a tower of finite cyclic cover with exponential torsion homology growth is asymptotically one. Our argument also generalizes a result of Dunfield and Thurston [3] where they showed the probability of a random 3-manifold of genus 2 having an abelian cover with positive Betti number is asymptotically zero. In [1] we obtain the same result for all genera.

[1] H. Baik, D. Bauer, I. Gekhtman, U. Hamenstädt, S. Hensel, T. Kastenholz, B. Petri and D. Valenzuela, Exponential torsion growth for random 3-manifolds, to appear in Int. Math. Res. Not. IMRN.
[2] N. Bergeron and A. Venkatesh, The asymptotic growth of torsion homology for arithmetic groups. J. Inst. Math. Jussieu 12 (2013), no. 2, 391–447.
[3] N. Dunfield and W. Thurston, Finite covers of random 3-manifolds, Inventiones Mathematicae 166 (2006), pp. 457–521.

Geometric limits of Kleinian groups

클리이니안 군 (쌍곡 3차원다양체의 기본군)의 기하적 수렴에 대한 연구

A natural topology on the space of Kleinian groups is given by Hausdorff distance, called Geometric topology or Chabauty topology. It is an interesting question to understand the global shape of the closure of the space of a given class of Kleinian groups in geometric topology. For instance, the closure of the space of once-punctured torus groups are conjectured to coincide with the set of primitive stable type-preserving representations [4]. As a preliminary case, I have worked out explicitly with L. Clavier in the case of closed abelian subgroups of PSL2(R) [1] or PSL2(C) [2]. One of the main results of [1] is a tool which can reduce a problem of convergence in Chabauty topology in a given space to a problem of convergence in Chabauty topology in a better understood space like C*. The Chabauty topology on C* is described in [3].

[1] H. Baik and L. Clavier, The Space of Geometric Limits of One-generator Closed Subgroups of PSL(2, R), Algebr. Geom. Topol., Vol 13, Issue 1 (2013), pp. 549-576.
[2] H. Baik and L. Clavier, The Space of Geometric Limits of Abelian Subgroups of PSL(2,C), Hiroshima Math. J. (2016), Volume 46, Number 1, 1-36.
[3] H. Baik and L. Clavier, Introduction to Chabauty topology and Pictures of the Chabauty space of C∗, arXiv:1209.0221.
[4] Y. Minsky, On dynamics of Out(Fn) on PSL2(C) characters, Isr. J. Math. (2013) 193: 47.