On Morava K-theories of an S-arithmetic Group

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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Mod p Arithmetic, Group Theory and Cryptography

An Invitation to Modern Number Theory ◽

10.2307/j.ctv131bw89.6 ◽

2020 ◽

pp. 3-28

Keyword(s):

Group Theory ◽

Arithmetic Group ◽

Mod P

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LINK COMPLEMENTS ARISING FROM ARITHMETIC GROUP ACTIONS

International Journal of Mathematics ◽

10.1142/s0129167x95000109 ◽

1995 ◽

Vol 06 (03) ◽

pp. 337-370 ◽

Cited By ~ 16

Author(s):

FRITZ GRUNEWALD ◽

ULRICH HIRSCH

Keyword(s):

Quotient Space ◽

Free Subgroup ◽

Arithmetic Group ◽

Hyperbolic Structure ◽

3 Dimensional ◽

Ring Of Integers ◽

Hyperbolic Volume ◽

Link Complements ◽

Torsion Free Subgroup ◽

Torsion Free

Let [Formula: see text] be a torsion-free subgroup acting discontinuously on 3-dimensional hyperbolic space [Formula: see text]. Assume further that Γ\ℍ3 has finite hyperbolic volume. The quotient-space Γ\ℍ3 is then a 3-manifold which can be compactified by the addition of finitely many 2-tori. This paper discusses a procedure which decides whether Γ\ℍ3 is homeomorphic to the complement of a link in S3. We apply our procedure to subgroups of low index in [Formula: see text], where [Formula: see text] is the ring of integers in [Formula: see text]. As a result we find new link complements having a complete hyperbolic structure coming from an arithmetic group. Finally we prove that up to conjugacy there are only finitely many commensurability classes of arithmetic subgroups [Formula: see text] so that Γ\ℍ3 is homeomorphic to the complement of a link in S3.

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