An arithmetic group associated with a Pisot unit, and its symbolic-dynamical representation

International audience Let T=T(A,D) be a self-affine tile in \reals^n defined by an integral expanding matrix A and a digit set D. In connection with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers \0,1,..., |det(A)|-1\. It is shown that in \reals^3 and \reals^4, for any integral expanding matrix A, T(A,D) is connected. We also study the connectedness of Pisot dual tilings which play an important role in the study of β -expansion, substitution and symbolic dynamical system. It is shown that each tile generated by a Pisot unit of degree 3 is arcwise connected. This is naturally expected since the digit set consists of consecutive integers as above. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4. Also we give a simple necessary and sufficient condition for the connectedness of the Pisot dual tiles of degree 4. As a byproduct, a complete classification of the β -expansion of 1 for quartic Pisot units is given.

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Arithmetic group symmetry and finiteness properties of Torelli groups

Annals of Mathematics ◽

10.4007/annals.2013.177.2.1 ◽

2013 ◽

Vol 177 (2) ◽

pp. 395-423 ◽

Cited By ~ 1

Author(s):

Alexandru Dimca ◽

Stefan Papadima

Keyword(s):

Group Symmetry ◽

Arithmetic Group ◽

Finiteness Properties

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Arithmetic of arithmetic Coxeter groups

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1809537115 ◽

2018 ◽

Vol 116 (2) ◽

pp. 442-449 ◽

Cited By ~ 2

Author(s):

Suzana Milea ◽

Christopher D. Shelley ◽

Martin H. Weissman

Keyword(s):

Coxeter Group ◽

Quadratic Forms ◽

Coxeter Groups ◽

Diophantine Equations ◽

Geometric Method ◽

Arithmetic Group ◽

Binary Quadratic Forms ◽

Pell’S Equation

In the 1990s, J. H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the “topograph,” Conway revisited the reduction of BQFs and the solution of quadratic Diophantine equations such as Pell’s equation. It appears that the crux of his method is the coincidence between the arithmetic group PGL2(Z) and the Coxeter group of type (3,∞). There are many arithmetic Coxeter groups, and each may have unforeseen applications to arithmetic. We introduce Conway’s topograph and generalizations to other arithmetic Coxeter groups. This includes a study of “arithmetic flags” and variants of binary quadratic forms.

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UNIVERSAL 2-BRIDGE KNOT AND LINK ORBIFOLDS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s021821659300009x ◽

1993 ◽

Vol 02 (02) ◽

pp. 141-148 ◽

Cited By ~ 2

Author(s):

HUGH M. HILDEN ◽

MARIA TERESA LOZANO ◽

JOSÉ MARIA MONTESINOS-AMILIBIA

Keyword(s):

Fundamental Group ◽

Finite Index ◽

Discrete Group ◽

Prime Numbers ◽

Arithmetic Group ◽

Singular Set ◽

Mod P

Let (p/q, n) be the orbifold with cyclic isotropy of order n and with singular set the 2-bridge knot or link p/q where p and q are relatively prime numbers, q is odd, q is less than p, and q is not congruent to ±1 mod p (i.e. p/q is any non toroidal 2-bridge knot or link). We show that the orbifold fundamental group π1(p/q, n) is universal for n any multiple of 12. This means that if Γ is any such group, it can be thought of as a discrete group of hyperbolic isometries of hyperbolic 3-space ℍ3, and then, given any closed, oriented 3-manifold M, there exists a subgroup of finite index G of Γ such that M is homeomorphic to G\ℍ3. Since we have shown elsewhere that the group π1(5/3, 12) is an arithmetic group, it follows that there exists an orbifold, namely (5/3, 12), whose singular set is a knot, the figure eight, and whose fundamental group is both arithmetic and universal.

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DERIVED HECKE ALGEBRA AND COHOMOLOGY OF ARITHMETIC GROUPS

Forum of Mathematics Pi ◽

10.1017/fmp.2019.6 ◽

2019 ◽

Vol 7 ◽

Author(s):

AKSHAY VENKATESH

Keyword(s):

Hecke Algebra ◽

Galois Cohomology ◽

Arithmetic Group ◽

Arithmetic Groups ◽

Cohomology Groups ◽

Single Degree ◽

Cohomology Of Arithmetic Groups ◽

Favorable Conditions

We describe a graded extension of the usual Hecke algebra: it acts in a graded fashion on the cohomology of an arithmetic group $\unicode[STIX]{x1D6E4}$ . Under favorable conditions, the cohomology is freely generated in a single degree over this graded Hecke algebra. From this construction we extract an action of certain $p$ -adic Galois cohomology groups on $H^{\ast }(\unicode[STIX]{x1D6E4},\mathbf{Q}_{p})$ , and formulate the central conjecture: the motivic $\mathbf{Q}$ -lattice inside these Galois cohomology groups preserves $H^{\ast }(\unicode[STIX]{x1D6E4},\mathbf{Q})$ .

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