We consider the pseudo-Anosov elements of the mapping class group of a surface of genus g that fix a rank k subgroup of the first homology of the surface. We show that the smallest entropy among these is comparable to (k + 1)/g. This interpolates between results of Penner and of Farb and the second and third authors, who treated the cases of k = 0 and k = 2g, respectively, and answers a questio...

The SL(2, R)/U(1) coset model, with U(1) an element of the third conjugacy class of SL(2, R) subgroups, is considered. The resulting theory is seen to collapse to a one dimensional field theory of Liouville. Then the 2 dimensional black hole SL(2, R)/U(1), with U(1) a non-compact subgroup boosted by a Lorentz transformation, is considered. In the limit of high boost, the resulting black hole is...

Let Σ be a compact orientable surface with genus g and n boundary components B = (B1, . . . , Bn). Let c = (c1, . . . , cn) ∈ [−2, 2]n. Then the mapping class group MCG of Σ acts on the relative SU(2)-character variety XC := HomC(π,SU(2))/SU(2), comprising conjugacy classes of representations ρ with tr(ρ(Bi)) = ci. This action preserves a symplectic structure on the smooth part of XC , and the ...

By algebraic group theory, there is a map from the semisimple conjugacy classes of a nite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. We conjecture that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of ...

Abstract We prove that the number of conjugacy classes a finite group G consisting elements odd order, is larger than or equal to for normaliser Sylow 2-subgroup . This predicted by Alperin Weight Conjecture.

We show that any infinite algebraic subgroup of the plane Cremona group over a perfect field is contained in maximal group. classify groups, and their subgroups rational points, up to conjugacy by birational map.

We prove the Arad–Herzog conjecture for various families of finite simple groups — if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad–Herzog co...

In this paper we study conjugacy separability of subdirect products of two free (or hyperbolic) groups. We establish necessary and sufficient criteria and apply them to fibre products to produce a finitely presented group G1 in which all finite index subgroups are conjugacy separable, but which has an index 2 overgroup that is not conjugacy separable. Conversely, we construct a finitely present...