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European Mathematical Society Publishing House
2024-03-28 17:37:07
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=CMH&vol=92&iss=1&update_since=2024-03-28
Commentarii Mathematici Helvetici
Comment. Math. Helv.
CMH
0010-2571
1420-8946
General
10.4171/CMH
http://www.ems-ph.org/doi/10.4171/CMH
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
92
2017
1
Word length statistics for Teichmüller geodesics and singularity of harmonic measure
Vaibhav
Gadre
University of Glasgow, GLASGOW, UNITED KINGDOM
Joseph
Maher
CUNY College of Staten Island, STATEN ISLAND, UNITED STATES
Giulio
Tiozzo
University of Toronto, TORONTO, ONTARIO, CANADA
Random walks, mapping class group, Teichmüller flow, harmonic measure, cusp excursion
Given a measure on the Thurston boundary of Teichmüller space, one can pick a geodesic ray joining some basepoint to a randomly chosen point on the boundary. Different choices of measures may yield typical geodesics with different geometric properties. In particular, we consider two families of measures: the ones which belong to the Lebesgue or visual measure class, and harmonic measures for random walks on the mapping class group generated by a distribution with finite first moment in the word metric. We consider the word length of approximating mapping class group elements along a geodesic ray, and prove that this quantity grows superlinearly in time along almost all geodesics with respect to Lebesgue measure, while along almost all geodesics with respect to harmonic measure the growth is linear. As a corollary, the harmonic and Lebesgue measures are mutually singular. We also prove a similar result for the ratio between the word metric and the relative metric (i.e. the induced metric on the curve complex).
Functions of a complex variable
Several complex variables and analytic spaces
Probability theory and stochastic processes
1
36
10.4171/CMH/404
http://www.ems-ph.org/doi/10.4171/CMH/404
A classification of torsors over Laurent polynomial rings
Vladimir
Chernousov
University of Alberta, EDMONTON, CANADA
Philippe
Gille
Université Claude Bernard Lyon 1, VILLEURBANNE CEDEX, FRANCE
Arturo
Pianzola
University of Alberta, EDMONTON, CANADA
Reductive group scheme, torsor, multiloop algebra
Let $R_n$ be the ring of Laurent polynomials in $n$ variables over a field $k$ of characteristic zero and let $K_n$ be its fraction field. Given a linear algebraic $k$-group $G$, we show that a $K_n$-torsor under $G$ which is unramified with respect to $X$ = Spec$(R_n)$ extends to a unique toral $R_n$-torsor under $G$. This result, in turn, allows us to classify all $G$-torsors over $R_n$.
Algebraic geometry
Number theory
Nonassociative rings and algebras
Group theory and generalizations
37
55
10.4171/CMH/405
http://www.ems-ph.org/doi/10.4171/CMH/405
Fundamental domains and generators for lattice Veech groups
Ronen
Mukamel
Rice University, HOUSTON, UNITED STATES
Riemann surfaces, Teichmüller theory, Veech groups
The moduli space $Q \mathcal M_g$ of non-zero genus $g$ quadratic differentials has a natural action of $G=\mathrm {GL}_2^+(\mathbb R)$ / $\langle ± \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ $\rangle$. The Veech group PSL$(X,q)$ is the stabilizer of $(X,q) \in Q \mathcal M_g$ in $G$. We describe a new algorithm for finding elements of PSL$(X,q)$ which, for lattice Veech groups, can be used to compute a fundamental domain and generators. Using our algorithm, we give the first explicit examples of generators and fundamental domains for non-arithmetic Veech groups where the genus of $\mathbb H$ / PSL$(X,q)$ is greater than zero.
Several complex variables and analytic spaces
Functions of a complex variable
57
83
10.4171/CMH/406
http://www.ems-ph.org/doi/10.4171/CMH/406
On a modular Fermat equation
Jonathan
Pila
University of Oxford, OXFORD, UNITED KINGDOM
Zilber–Pink conjecture, Fermat equation, o-minimality
We study some diophantine problems suggested by the analogy between multiplicative groups and powers of the modular curve in problems of “unlikely intersections”. We prove a special case of the Zilber–Pink conjecture for curves.
Number theory
Mathematical logic and foundations
85
103
10.4171/CMH/407
http://www.ems-ph.org/doi/10.4171/CMH/407
Free loci of matrix pencils and domains of noncommutative rational functions
Igor
Klep
The University of Auckland, AUCKLAND, NEW ZEALAND
Jurij
Volčič
The University of Auckland, AUCKLAND, NEW ZEALAND
Linear pencil, noncommutative rational function, realization theory, free locus, real algebraic geometry, hyperbolic polynomial, Kippenhahn’s conjecture
Consider a monic linear pencil $L(x)=I-A_1x_1-\cdots-A_gx_g$ whose coefficients~$A_j$ are $d\times d$ matrices. It is naturally evaluated at $g$-tuples of matrices $X$ using the Kronecker tensor product, which gives rise to its free locus $\mathcal (L)=\{X:\det L(X)=0\}$. In this article it is shown that the algebras $\mathcal A$ and $\widetilde{\mathcal A}$ generated by the coefficients of two linear pencils $L$ and $\widetilde{L}$, respectively, with equal free loci are isomorphic up to radical, i.e., $\mathcal A/\mathrm {rad}\mathcal A\cong \widetilde{\mathcal A}/\mathrm {rad}\widetilde{\mathcal A}$. Furthermore, $\mathcal (L)\subseteq \mathcal(\widetilde{L})$ if and only if the natural map sending the coefficients of $\widetilde{L}$ to the coefficients of $L$ induces a homomorphism $\widetilde{\mathcal A}/\mathrm {rad}\widetilde{\mathcal A}\to \mathcal A/\mathrm {rad}\mathcal A$. Since linear pencils are a key ingredient in studying noncommutative rational functions via realization theory, the above results lead to a characterization of all noncommutative rational functions with a given domain. Finally, a quantum version of Kippenhahn's conjecture on linear pencils is formulated and proved: if hermitian matrices $A_1,\dots,A_g$ generate $M_d(\mathbb C)$ as an algebra, then there exist hermitian matrices $X_1,\dots,X_g$ such that $\sum_iA_i\otimes X_i$ has a simple eigenvalue.
Linear and multilinear algebra; matrix theory
Algebraic geometry
Associative rings and algebras
Real functions
105
130
10.4171/CMH/408
http://www.ems-ph.org/doi/10.4171/CMH/408
Ergodic components of partially hyperbolic systems
Andy
Hammerlindl
Monash University, CLAYTON, VIC, AUSTRALIA
Partial hyperbolicity, ergodic decompositions, accessibility classes
This paper gives a complete classification of the possible ergodic decompositions for certain open families of volume-preserving partially hyperbolic diffeomorphisms. These families include systems with compact center leaves and perturbations of Anosov flows under conditions on the dimensions of the invariant subbundles. The paper further shows that the non-open accessibility classes form a $C^1$ lamination and gives results about the accessibility classes of non-volume-preserving systems.
Dynamical systems and ergodic theory
Manifolds and cell complexes
131
184
10.4171/CMH/409
http://www.ems-ph.org/doi/10.4171/CMH/409
Differentiability of integrable measurable cocycles between nilpotent groups
Michael
Cantrell
University of Illinois at Chicago, CHICAGO, UNITED STATES
Cocycle ergodic theorems, integrable measure equivalence, nilpotent groups, asymptotic cones, Pansu derivative
We prove an analog for integrable measurable cocycles of Pansu’s differentiation theorem for Lipschitz maps between Carnot–Carathéodory spaces. This yields an alternative, ergodic theoretic proof of Pansu’s quasi-isometric rigidity theorem for nilpotent groups, answers a question of Tim Austin regarding integrable measure equivalence between nilpotent groups, and gives an independent proof and strengthening of Austin’s result that integrable measure equivalent nilpotent groups have bi-Lipschitz asymptotic cones. Our main tools are a nilpotent-valued cocycle ergodic theorem and a Poincaré recurrence lemma for nilpotent groups.
Group theory and generalizations
Measure and integration
Dynamical systems and ergodic theory
185
213
10.4171/CMH/410
http://www.ems-ph.org/doi/10.4171/CMH/410