Perhaps it's worth mentioning that Property (T) seems to repel certain strengthenings of residual finiteness. An open question of Long and Reid asks:

Is there an infinite finitely generated group which has LERF and has Property (T)?

Recall that LERF stands for Locally Extended Residually Finite, and means that every finitely generated subgroup is closed in the profinite topology. (Residual finiteness means the trivial subgroup is closed)

Long and Reid's question may have a positive answer, but the point of it is that such groups seem in any case to be extremely rare and/or difficult to construct.

Perhaps it's worth mentioning that Property (T) seems to repel certain strengthenings of residual finiteness. An open question of Long and Reid asks:

Is there an infinite finitely generated group which is LERF and has Property (T)?

Recall that LERF stands for Locally Extended Residually Finite, and means that every finitely generated subgroup is closed in the profinite topology. (Residual finiteness means the trivial subgroup is closed)

Long and Reid's question may have a positive answer, but the point of it is that such groups seem in any case to be extremely rare and/or difficult to construct.

Perhaps it's worth mentioning that Property (T) seems to repel certain strengthenings of residual finiteness. An open question of Long and Reid asks:

Is there a finitely generated group which has LERF and has Property (T)?

Recall that LERF stands for Locally Extended Residually Finite, and means that every finitely generated subgroup is closed in the profinite topology. (Residual finiteness means the trivial subgroup is closed)

Long and Reid's question may have a positive answer, but the point of it is that such groups seem in any case to be extremely rare and/or difficult to construct.

Perhaps it's worth mentioning that Property (T) seems to repel certain strengthenings of residual finiteness. An open question of Long and Reid asks:

Is there an infinite finitely generated group which has LERF and has Property (T)?

Recall that LERF stands for Locally Extended Residually Finite, and means that every finitely generated subgroup is closed in the profinite topology. (Residual finiteness means the trivial subgroup is closed)

Long and Reid's question may have a positive answer, but the point of it is that such groups seem in any case to be extremely rare and/or difficult to construct.